|sAN     FRANCISCO     STATE     NORMAL     SCHOOL 

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I^9Q     I  UC-NRLF 

^^^  B  ^  SIS  ^si 


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How  to  Teach  the  Critical 
Difficulties  of  Arithmetic 

to  accompany  the 

CALIFORNIA  STATE  SERIES  ADVANCED 
ARITHMETIC 


•  /  r.i 


For  Normal  Students  and  Experienced  Teachers 

By   MARY  A.  WARD 

Supervisor  of  the  Teaching  of  Arithmetic 


California 
1  State  Printing  Office 

,  15484  1915 


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TABLE  OF  CONTENTS. 


Page. 

INTRODUCTION 1 

CRITICAL,      DIFFICULTIES      IN      THE      ADDITION      AND      SUBTRACTION      OF 

FRACTIONS  AND  HOW  TO  TEACH  THEM 2 

CRITICAL  DIFFICULTIES  IN  THE  MULTIPLICATION  AND  DIVISION  OF  FRAC- 
TIONS AND  HOW  TO  TEACH  THEM 9 

MISCELLANEOUS  QUESTIONS  ON  FRACTIONS 11 

TESTS    IN    ADDITION,    SUBTRACTION,    MULTIPLICATION    AND    DIVISION    OF 

FRACTIONS 12 

CRITICAL   DIFFICULTIES    IN   THE   ADDITION   AND   SUBTRACTION    OF   DECI- 
MALS AND  HOW  TO  TEACH  THEM 13 

CRITICAL  DIFFICULTIES  IN  THE  MULTIPLICATION  AND  DIVISION  OF  DECI- 
MALS AND  HOW  TO  TEACH  THEM 15 

MISCELLANEOUS  QUESTIONS  ON  DECIMALS 19 

TESTS  IN   THE  ADDITION,    SUBTRACTION,   MULTIPLICATION  AND   DIVISION 

OF   DECIMALS    21 

TEST  QUESTIONS  ON  AVERAGE,   RATIO,    TABLES,   BILLS  AND  RECEIPTS 21 

SQUARE   MEASURE 22 

CUBIC    MEASURE 25 

SUGGESTIONS  FOR  TEACHING  EACH  CASE  IN  PERCENTAGE 26 

TEST  ON  FORMAL  WORK  OF  THE  THREE    CASES    OF    PERCENTAGE 2S 

TEST   ON   PROBLEM  WORK   OF   THE   THREE     CASES    OF    PERCENTAGE 29 

TEST   ON   COMMISSION,   INSURANCE,   TAXES,  DUTIES,  TRADE  DISCOUNT  AND 

INTEREST    31 

COMPLETE   REVIEW   OF   MENSURATION 33 

CLASSIFICATION  OF  PROBLEMS  OF  PAGES  18-160  OF  TEXT 37 


306779 


I 


HOW  TO  TEACH  THE  CRITICAL  DIFFICULTIES  OF 

ARITHMETIC. 


INTRODUCTION. 

The  helps  and  suggestions  given  in  this  bulletin  for  teaching  each  of  the 
topics  of  the  State  Series  Advanced  Arithmetic  may  be  applied  to  the  work 
of  any  arithmetic  text.  Teachers  will  find  the  miscellaneous  questions  and 
tests  given  at  the  end  of  each  topic  useful  in  many  ways.  They  may  be  used 
as  thorough  reviews  at  the  beginning  of  a  new  term  or  as  tests  for  new 
pupils  who  enter  late  in  the  term.  Such  tests  should  help  to  determine  the 
points  in  which  a  pupil's  knowledge  is  deficient.  The  classification  of  the 
problems  of  pages  18  through  165  of  the  State  text  will  be  of  much  service 
to  the  teacher  who  is  using  this  portion  of  the  textbook. 

High  school  students  desiring  to  enter  the  Normal  School,  will  find  that 
the  bulletin  outlines  the  scope  and  type  of  the  arithmetic  work  with  which 
students  must  be  familiar  before  they  can  be  entrusted  with  the  teaching  of 
this  subject  in  the  classes  of  the  training  school.  Students  will  be  expected 
to  be  familiar  with  the  tests,  suggestions  and  methods  of  work  given  in  this 
book. 

Besides  the  subjects  treated  in  this  outline,  normal  students  are  expected 
to  be  familiar  with  the  following: 

1.  The  system  of  addition  combinations  as  given  in  chapter  three  of  the 

State  primary  text. 

2.  The  method  of  finding  the  quotient  figures  in  long  division.    (Advanced 

State  Text,  pages  63-67.) 

3.  Additive  method  of  subtraction.     (Advanced  Text,  pages  21-22.) 

MARY  A.  WARD. 


State  Normal  School,  San  Francisco, 
February,  1915. 


HOW  TO  TEACH  THE  CRITICAL  DIFFICULTIES  OF 

ARITHMETIC. 


COMMON  FRACTIONS. 

The  following  sections  summarize  the  various  types  of  problems  which 
should  be  taught  in  the  addition,  subtraction,  multiplication  and  division  of 
fractions.  The  suggestions  given  here  for  the  teaching  of  fractions  are 
intended  to  be  used  in  connection  with  the  indicated  sections  of  the  Cali- 
fornia State  Series  Advanced  Arithmetic,  but  may  be  used  in  connection 
with  the  fraction  work  of  any  arithmetic  text.  Normal  students  will  be 
required  to  work  problem^s  similar  to  the  types  given  here  and  to  use  the 
form  and  methods  of  work  indicated  in  this  outline.  In  order  to  follow 
the  references,  students  should  if.  possible  provide  themselves  with  a  (.-opy 
of  the  California  State  Series  Advanced  Arithmetic.  Students  should  be 
especially  familiar  with  the  work  of  Sections  26  and  27. 

1.  Fractions: 

1.  Teach  pupils  that  a  piece  and  a  fraction  of  an  object  are  one  and 
the  same. 

2.  That  the  number  of  equal  pieces  into  which  the  whole  quantity 
is  divided  determines  the  name  of  the  piece  or  fraction.  Ask — "If  a 
line  is  divided  into  4  equal  parts,  what  is  each  part  called  ?  If  divided 
into  2  equal  parts?     5  equal  parts?     etc."     (Text  page  90.) 

3.  Ask — -"How  many  fourths,  thirds,  tifths,  etc.  are  there  in  one 
whole?" 

4.  Teach  pupils  the  names  of  the  parts  of  a  fraction — numerator  and 
denominator.     (Text  page  93.) 

5.  Suggest  that  the  first  letter  of  the  word  down  is  d.  The  name 
of  the  term  which  is  down  or  below^  the  line,  denominator,  also  begins 
with  a  d. 

6.  Have  pupils  show  ^,  f,  f,  |-,  etc.,  of  lines,  rectangles,  circles.  Be 
careful  to  see  that  pupils  point  not  to  the  line  of  division,  but  to  the 
portion  of  the  line  or  surface  when  showing  ^,  f ,  ^,  etc.,  of  a  line,  rec- 
tangle, or  circle.     (Text  page  93,  numbers  6-7.) 

7.  Teach  pupils  to  distinguish  between  proper  and  improper  frac- 
tions and  mixed  numbers.  Do  not  teach  definitions  of  these  terms. 
(Text  page  94.) 

8.  Have  pupils  classify  each  of  the  following  as  proper  fractions,  im- 
proper fractions  or  as  mixed  numbers:  If,  |,  f,  |,  V,  If?  V-  Pupils 
often  fail  to  regard  ^,  |,  etc.,  as  improper  fractions. 

Write  a  miscellaneous  list  of  proper,  improper  fractions  and  mixed 
numbers  on  the  board.  Have  pupils  tell  whether  each  is  I  (improper), 
M  (mixed  number)  or  P  (proper).  This  form  of  drill  may  be  either 
oral  or  written. 

(2) 


2.  Changing  Improper  Fractions  to  Mixed  Numbers. 

Before  beginning  the  addition  of  fractions  of  like  denominators,  it  is 
necessary  that  pupils  know  how  to  change  improper  fractions  to  mixed 
numbers.  There  is  no  necessity  for  teaching-  pupils  to  change  mixed 
numbers  to  improper  fractions  at  this  time.  The  application  of  the 
drill  upon  changing  mixed  numbers  to  improper  fractions  can  not  be 
made  until  pupils  are  ready  for  work  in  the  multiplication  of  frac- 
tions. For  this  reason  the  work  of  Sections  119-120  of  text  should  be 
delayed  until  pupils  are  ready  to  begin  the  multiplication  of  fractions. 

3.  Change  Y-  to  a  Mixed  Number:   (Text  page  96.) 

a.  Teach  the  pupils  to  read  V  as  19  divided  by  3.  Emphasize  the 
fact  that  the  horizontal  line  is  the  sign  of  division.  Pupils  should 
read  y  in  two  ways— as  19  thirds  or  as  19  divided  by  3. 

1).  Ask:  What  is  the  answer  when  19  is  divided  by  3?  Nineteen 
thirds  equal  what?  Proceed  in  the  same  manner  with  other  improper 
fractions.  Do  not  teach  by  rule  the  changing  of  improper  fractions 
to  mixed  numbers. 

c.  When  pupils  can  readily  change  improper  fractions  to  mixed 
numbers,  show  the  reason  the  numerator  is  divided  by  the  denominator. 

Ask  :  How  many  thirds  are  there  in  one  whole "?  How  many  Avhole 
numbers  are  there  inf,  f,  f,  -y^,  i/,  etc.,  |,  |,  |,  V,  ¥,  ¥? 

d.  Require  pupils  to  change  mentally  all  small  improper  fractious 
to  mixed  numbers. 

4.  Reduction  to  Lower  Terms. 

Much  time  is  wasted  in  attempting  to  teach  pupils  to  find  the 
greatest  common  divisor  of  two  or  more  numbers.  That  the  reduc- 
tion of  fractions  to  lower  terms  may  be  taught  without  the  use  of 
such  terms  is  shown  by  the  following  suggestions  (Text  87-89)  : 

a.  Teach  that  the  even  numbers  are  0,  2,  4,  6,  8. 

&.  Ask :  Which  of  the  following  end  in  even  numbers  234,  40,  43, 
27,  22,  35,  19,  36? 

c.  Teach  pupils  to  recognize  which  of  the  following  fractions  have 
numerators  and  denominators  that  end  in  even  numbers. 


2.4      16      10       3  2.      .'5  7    pf^ 


d.  Teach  that  when  both  the  numerator  and  the  denominator  of  a 
fraction  end  in  even  numbers,  the  fraction  may  be  reduced  by  dividing 
both  numerator  and  denominator  by  2. 

Reduction  by  5:  Teach  that  when  the  numerator  and  the  denom- 
inator of  a  fraction  end  in  5,  or  0  and  5,  the  fraction  may  be  reduced 
by  dividing  both  numerator  and  denominator  of  the  fraction  by  5. 

Reduction  by  10:  Teach  that  when  both  the  numerator  and  the 
denominator  of  a  fraction  end  in  0,  the  fraction  may  be  reduced  by 
dividing  both  the  numerator  and  the  denominator  of  the  fraction  bv 

■    10.     Thus,  &{J=^. 


(3) 


7.  Reduction  by  3 :     Reduce  ^. 

a.  In  the  numerator,  12,  add  together  the  1  and  2.  The  answer  is  3. 
May  3  be  divided  by  3  ? 

&.  Add  together  the  3  and  9  in  the  denominator,  39.  The  answer 
is  12. 

c.  May  12  be  divided  by  3  ? 

d.  May  the  sum  of  the  numbers  in  the  numerator  of  ^|  be  divided 
by,3? 

e.  May  the  sum  of  the  numbers  in  the  denominator  of  ^|  be  divided 
by  3? 

/.  When  the  sum  of  the  numbers  in  the  numerator  and  the  sum  of 
the  numbers  in  the  denominator  of  a  fraction  are  divisible  by  3,  the 
original  fraction  is  divisible  by  3,  thus,  ||  =  j%. 

g.  Reduce  each  of  the  following  fractions  by  dividing  both  the 
numerator  and  the  denominator  of  the  fraction  by  3 : 

iiJLL     4  8.    IL     4  5. 
1728>^3'   7  25    72- 

li.  Give  much  drill  upon  the  reduction  of  small  fractions  such  as 
f  >  ¥j  I)  -Ni  li  6tc.  Pupils  should  be  able  to  reduce  such  types  men- 
tally without  any  hesitation. 

8.  Summary  of  Types  in  Reduction  to  Lower  Terms:     (Text  pages  101, 

104,  105.) 
<*•    Iff  J    ctI-       Divide  by  2  as  the  numerator  and  the  denominator  end 

in  even  numbers. 
&.   /oWj  f M-  Divide  by  5  as  the  numerator  and  the  denominator  end 
in  5,  or  0  and  5. 

c.  If  {],  -}f  Q.  Divide  by  10  as  the  numerator  and  the  denominator  end 

in  0. 

d.  Ill,  |gf .  Divide  by  3  as  the  sum  of  digits  of  numerator  and  the 

sum  of  digits  of  denominator  are  both  divisible  by  3. 

9.  Difficulties: 

1.  Besides  the  above  types,  pupils  often  fail  to  see  the  number  by 
which  fractions,  such  as  the  following,  may  be  reduced:  ^,  ^|,  \\,  ||, 

3  8- 

2.  Pupils  often  reduce  a  fraction  once  but  fail  to  reduce  it  to  its 
lowest  terms,  thus,  ^l  =  f  =  f . 

3.  Train  pupils  to  divide  both  numerator  and  denominator  of  frac- 
tions by  largest  number  possible. 

10.  Types  in  Addition  of  Fractions  of  Like  Denominations : 

For  drill  material  see  text  page  97. 

a.  h.  c.  d.  e, 

J-  ^  -  24-1-  6^ 

V  s:  a  "^X  'Jit 


2  1  X  A  6  A 

8  S  S  ^8  ^ 

41  141 


i  f  t  9  25 


11  2  42|  51 

(4) 


Teach  each  of  the  above  types  in  the  given  order.  Do  not  pass 
from  type  a  to  6  until  pupils  understand  type  a  perfectly.  Only  a 
few  problems  are  needed  to  teach  the  principles  of  each  type.  Follow 
closely  the  suggestions  given  under  the  following  sections : 

1.  Notice  that  in  type  a  the  answer  is  a  fraction  only.  Construct 
problems  to  fulfill  this  condition. 

2.  In  type  &  the  answer  ^-^  is  to  be  changed  mentally  to  the  mixed 
number,  1^  or  1^.  Nothing  is  put  upon  the  paper  except  the  answer 
in  its  simplest  form.  Notice  the  j^osition  of  the  |  in  the  answ^er. 
Where  is  the  whole  number,  2,  placed!  From  the  beginning  of  work 
with  this  type,  insist  upon  pupils  making  all  reductions  mentally. 

3.  In  type  c  pupils  must  be  shown  where  to  place  the  whole 
number  2. 

4.  In  type  d,  note  the  irregularity  of  the  column.  All  changing 
and  carrying  is  done  mentally. 

5.  In  type  e,  note  the  irregularity  of  the  column  and  the  position 
of  the  answer. 

6.  While  teaching  addition  of  fractions  of  like  denominators,  con- 
tinue drill  work  upon  the  reduction  of  fractions  to  lower  terms  and 
the  changing  of  improper  fractions  to  mixed  numbers. 

11.  Language  Forms  for  Addition  of  Fractions : 

Use  the  following  language  forms  in  dictating  exercises  in  addition : 
a.  Find  the  sum  of  4f ,  6^,  etc. 
h.  Add  the  following:    9|,  6^,  etc. 
c.  Add :    6f ,  81,  etc. 

12.  Addition  of  Fractions  of  Unlike  Denominators: 

Before  introducing  work  in  the  addition  of  fractions  of  unlike 
denominators,  pupils  should  be  able  —  n.  to  change  fractions  to  higher 
terms;  &.  to  find  the  smallest  common  denominator  for  several  unlike 
fractions. 

13.  Changing  Fractions  to  Higher  Terms :     (Text  pages  102-103.) 

a.  Teach  pupils  to  change  all  possible  fractions  to  6ths,  8ths,  12ths, 
ISths,  20ths,  24ths,  30ths,  36ths,  48ths,  60ths. 

h.  For  purposes  of  illustrating  the  changing  of  fractions  to  higher 
terms,  use  fractions  such  as  f,  f,  |  rather  than  \,  i,  or  ^.  Since  the 
numerators  of  the  latter  are  1,  they  are  exceptions  to  the  general  rule. 

14.  The  Smallest  Common  Denominator:     (Text  page  112,  col.  3.) 

fl.  Teach  pupils  that  1,  2,  3,  5,  7  are  prime  numbers.  That  is,  such 
numbers  are  divisible  only  by  themselves  and  one. 

6.  Teach  pupils  to  find  the  smallest  common  denominator  for  unlike 
fractions  by  the  following  method : 
Find  the  smallest  common  denominator  for  f,  §,  f,  |,  ^\^  f,  |. 
c.  Write  the  denominators  in  a  horizontal  line.     Thus, 
2  )^  — ^— jj— 8  — 12  — 7  — ^ 

2)] 4—    6  —  7 

2 3 7 

2X2X2X3X7  =  168. 

2—15484 


d.  Have  pupils  cross  out  all  the  small  denominators  which  are  con- 
tained an  even  number  of  times  in  the  larger  denominators.  Since 
4 — 3 — 6 — 2  are  contained  in  the  larger  number,  12,  they  are  crossed  out. 

e.  Have  pupils  divide  the  remaining  numbers  by  the  smallest  prime 
number  which  is  contained  "in  at  least  two  of  the  numbers  that  are 
left.  Continue  as  shoMTi  in  the  model.  Emphasize  the  fact  that  the 
divisor  must  be  contained  in  at  least  two  of  the  numbers  to  be  divided. 

f.  Pupils  should  be  perfectly  familiar  with  the  method  of  finding 
the  smallest  common  denominator  before  be^ning  the  addition  of 
fractions  of  unlike  denominators.  The  above  method  of  finding  the 
common  denominator  wdll  be  more  successful  with  the  ordinary  pupil 
than  the  inspection  method  given  on  pages  106  and  107  of  the 
Advanced  Text. 

15.  Difficulties  in  Teaching  Pupils  to  Find  the  Common  Denominator: 

a.  Pupils  sometimes  cross  out  both  the  large  and  small  numbers. 

Teach  that  the  larger  numbers  must  be  left  to  take  care  of  the  smaller 

numbers. 

&.  Pupils  often  fail  to  cross  out  small  numbers  when  it  is  possible 

to  do  so  in  the  second  or  third  line  of  the  work  thus :    Find  the  smallest 

common  denominator  for  #,   vV,  1,  4 


16. 


2) 

T2") 

.4- 

12- 

-8- 

-9 

2) 

6- 

-4- 

-9 

:1  — 

-2- 

-9 

2  X  2  X  2  X  9  =  72. 

c.  Pupils  often  waste  time  dividing  by  a  number  which  is  con- 
tained in  only  one  of  the  denominators.  In  the  above  model,  pupils 
might  divide  the  last  line  by  2  and  then  the  next  line  by  9,  which 
would  be  a  great  waste  of  time.  Insist  upon  pupils  dividing  by  the 
smallest  number  which  is  contained  in  at  least  two  of  the  denomi- 
nators. 

d.  In  teaching  the  above  work,  always  dictate  the  unlike  fractions 
which  are  to  be  added.  Do  this  so  that  pupils  may  see  that  they  are 
learning  to  find  the  smallest  common  denominator  of  several  numbers 
solelj^  for  the  purpose  of  adding  unlike  fractions. 

e.  Be  certain  that  the  fractions  which  you  dictate  for  the  above  work 
have  denominators  which  illustrate  the  several  steps  of  finding  the 
smallest  common  denominator. 

Forms  for  Addition  of  Fractions  of  Unlike  Denominators. 

All  that  remains  to  be  taught  in  the  addition  of  fractions  of  unlike 
denominators  is  the  form  of  the  work. 
a. 

J.J  —  ,,^^  j_j 

81  =  84^  8| 

AS.  —  4.   9  4.3 


1). 


^4 

2T 


A 


29^'^ 
-3  —  2 


29  y^ 

-8  =  24  S.  C.  D. 
(6) 


a.  Pupils  who  are  learning  the  addition  of  fractions  should  use  the 
form  of  model  a. 

h.  Model  1)  is  shorter  and  may  be  used  after  the  pupils  have  passed 
the  work  of  multiplication  of  fractions,  or  sooner  if  the  teacher  feels 
that  the  pupils  know  what  they  are  doing. 

c.  The  M'ork  of  finding  the  new  denominator  should  always  appear  on 
the  pupil's  paper.  In  this  way  a  pupil's  knowledge  of  this  principle 
vciSiY  be  tested. 

d.  Notice  that  the  answer  to  the  addition  problem  of  the  above  model 
is  written  on  the  paper  only  in  its  simplest  form.  Train  pupils  to  do  as 
much  as  possible  of  the  reduction  mentally. 

e.  Train  pupils  to  reduce  fractions  of  the  given  problem  to  lowest 
terms  before  attempting  to  find  the  common  denominator  of  the  frac- 
tions.    In  this  way  much  useless  work  is  saved. 

Find  the  sum  of  -/-j,  -{^i^  i>  li  f)  iV-  First  reduce  the  fractious  to 
lower  terms.      The  problem  then  becomes :      Find  the  sum  of  ^,  f , 


2^11 
3>    4>    4)    4- 

17.  Difficulties  Pound  in  the  Teaching  of  Addition  of  Fractions. 

a.  The  changing  of  fractions  to  higher  terms. 

b.  Finding  the  smallest  common  denominator  for  several  unlike 
fractions. 

c.  Changing  mentally  improper  fractions  to  mixed  numbers  and 
reducing  fractions  to  lower  terms. 

d.  Failure  of  pupils  to  use  the  correct  form  when  adding  fractions  of 
unlike  denominators.  Teach  pupils  why  the  following  form  is  incor- 
rect:     2|=^V 

e.  Be  careful  to  see  that  the  problems  you  use  for  drill  work  illustrate 
the  principle  upon  which  you  are  drilling.  In  teaching  the  method  of 
finding  smallest  common  denominator,  do  not  use  a  problem  such  as 

/.  Do  not  introduce  the  subtraction  of  fractions  until  pupils  have  a 
thorough  knowledge  of  each  of  the  above  difficulties. 

SUBTRACTION  OF  FRACTIONS. 

18.  Types  in  Subtraction  of  Fractions  Which  Involve  No  Carrying: 

For  review  work  after  types  a,  h,  c,  and  d  have  been  taught,  use  the 
material  on  pages  98-99  of  text. 

Type  a.  Larger  Fraction  in  Minuend :  295f  ^ 

-  m     -I 

Type  1).  Fractions  in  Minuend  Only:  249f 

—  18 

Type  c.  Fraction  Only  Subtracted  From  Mixed  Number: 

832X 
a 


Type  d.  Answer  a  Whole  Number  Only  :        2H 


18  4  4 

-■"O  9  9 

(7) 


Suggestions : 

1.  Teach  each  of  the  above  types  one  at  a  time.  Do  cot  attempt  to 
make  pupils  conscious  of  which  types  they  are  learning. 

2.  Insist  upon  pupils  making  all  reductions  mentally. 

3.  Teach  pupils  to  prove  their  answers  to  all  problems  in  the  sub- 
traction of  fractions. 

4.  Be  certain  that  pupils  can  work  problems  of  the  above  types  when 
arranged  in  a  miscellaneous  order. 

5.  For  extra  drill  work,  have  pupils  make  up  and  solve  problems  of 
their  own  which  involve  the  different  tj^pes.  Such  problems  are  a  good 
test  of  pupil's  knowledge  of  the  work  covered. 

19.  Types  in  Subtraction  of  Fractions  That  Involve  Carrying: 

a.  Fraction  in  Subtrahend  Only:  1018  1018 

—     27#      —         # 


1.  Teach  the  above  type  b.y  borrowing  method. 

2.  Do  uot  permit  pupil  to  put  changes  on  paper. 
h.  Smaller  Fraction  in  Minuend : 

32i  321 

-19|  -     f 

1.  Teach  the  above  type  by  the  borrowing  method. 

2.  Train  pupils  to  make  all  changes  mentally. 

3.  As  this  is  the  most  difficult  type  in  the  subtraction  of  fractions, 
pupils  should  receive  a  great  deal  of  drill  upon  it. 

4.  Teach  pupils  to  prove  all  answers  to  subtraction  problems. 

20.  Language  Forms: 

a.  When  pupils  can  work  readily  any  of  the  above  types  in  the  sub- 
traction of  fractions,  the  following  language  forms  should  be  taught 
and  applied : 

a.  Find  the  difference  betw^een  18|  and  44^.  The  difficulty  lies  in 
the  fact  that  the  smaller  number  is  dictated  first.  Pupils  are  inclined 
to  place  as  the  minuend  the  first  number  stated  in  the  problem. 

1).  Find  the  difference  between  32f  and  14^. 

c.  Subtract  9J  from  46^. 

d.  What  must  be  added  to  24^  to  give  32f  ? 

21.  Subtraction  of  Fractions  of  Unlike  Denominators : 

For  drill  material  see  text  page  110. 

3  i 3  .5 

f  8  'i'6 

1   8 


241  =  24^ 
-  6|=   f:-8_  _4 


17  ~  11 

a.  Notice  tlie  form  of  the  work. 

h.  Use  the  language  forms  given  under  the  subtraction  of  fractions  of 
like  denominators.     See  section  20. 

c.  Teach  the  reason  this  form  of  work  is  incorrect :     2'4  =  .%. 


(8) 


a. 

1X4-  ? 

h. 

f  X  1  =  ? 

c. 

^XA=? 

MULTIPLICATION  OF  FRACTIONS. 

The  nmltiplieation  of  fractions  is  to  be  taught  by  the  cancellation 
method.     The  following  is  the  order  in  which  the  ditferent  types  are  to 
be  taught :     For  drill  material  see  advanced  text  page  114,  section  152. 
(No  cancellation.) 
(Cancellation  cione  diagonally.) 
(Cancellation   done   vertically.     Emphasize   the   fact 
that  cancelling  is  simply  dividing  a  number  above 
the  line  and  a  number  below  the  line  by  the  same 
number.     Show  that  for  this  reason  we  never  cancel 
horizontally.)     Text  pages  119,  120,  121. 

d.  24  X  ^=  ?      (Teach  that  24  is  to  be  considered  as  the  fraction  \*-. 

Pupils  often  fail  upon  this  type  through  regarding 
24  as  the  denominator  of  a  fraction.)  Text  pages 
115-116. 

e.  f  X  12  =  ?     (The  same  difficulty  appears  in  this  type  as  in  Type  d.) 

22.  Changing  Mixed  Numbers  to  Improper  Fractions: 

It  is  at  this  point  that  pupils  should  be  taught  to  change  mixed  num- 
bers to  improper  fractions.     (See  text  page  95.) 
Change  2f  to  an  improper  fraction. 

a.  First,  teach  pupils  to  change  2f  to  an  improper  fraction  simply 
as  a  mechanical  operation  :     4X2^8  +  3  =  \i-. 

J).  Later  show  pupils  why  this  is  done.  Ask.  "How  many  fourths 
are  there  in  one  whole"     ( f )     Two  wholes?     Three  wholes?     etc." 

c.  In  2f  we  have  the  whole  number  2  or  f  and  f  more  or  y . 

d.  Pupils  should  be  able  to  readily  change  any  mixed  number  to  an 
improper  fraction  before  the  following  types  in  the  multiplication  of 
fractions  are  introduced. 

/.  2-j  X  f  =  ?       (Pupils  often  attempt  to  cancel  before  changing  the 
mixed  number  to  an  improper  fraction.     Use  small 
numbers.) 
g.  2|  X  24=  ?     (There  are  two  difficulties  to  be  noted  in  this  type. 
1.  Changing  the  mixed  number  to  an  improper 
fraction.     2.  Regarding  24  as  a  correct  fraction.) 
h.     I  X  2f  =  ?     (The  same  difficulty  appears  in  this  type  as  in  Type  g.) 
i.  5§  X  4f  =  ?     (Be  careful  while  teaching  the  principle  of  this  type 
to  use  small  numbers.) 
Suggestions : 

1.  Drill  upon  each  of  the  above  types  until  it  has  been  mastered.  Do 
not  be  in  too  great  a  hurry  to  introduce  a  new  type.  As  each  new  type 
is  mastered,  review  in  a  miscellaneous  order  all  previous  types. 

2.  There  is  no  necessity  for  using  large  numbers  when  teaching  the 
multiplication  of  fractions.  No  number  larger  than  25  should  be 
used. 

3.  Insist  upon  all  answers  being  reduced  to  lowest  terms. 


(9) 


23.  Language  Forms  for  Multiplication  of  Fractions: 

a.  Find  the  product  of  |  and  ^. 
l.  Multiply  12f  by  |. 

c.  Find  y  of  21.     (Pupils  should  be  able  to  illustrate  this  type  with 

the  aid  of  a  diagram.)     Text  page  138,  Sections  179  and  181. 

d.  Find  f  of  f. 

e.  What  is  |  of  6f  ? 

DIVISION  OF  FEACTIONS. 

1.  Have  pupils  read  problems  in  the  division  of  fractions  and  tell 
which  fraction  is  the  divisor  and  which  the  dividend. 

2.  Emphasize  the  fact  that  it  is  the  divisor  which  is  inverted. 

3.  Emphasize  the  fact  that  fractions  may  be  cancelled  only  when 
the}'  appear  with  multiplication  sign  between  them. 

4.  Pupils  should  be  taught  to  prove  their  answers  to  division 
problems. 

5.  The  following  is  a  list  of  the  types  of  problems  in  the  division  of 
fractions.  The  types  are  listed  in  the  order  in  which  they  are  to  be 
taught. 

a.  f--f=? 
c.  24  ^  f  =  ? 
e.  6  -f-  4§  =  ? 

g.  2f-|=? 

After  each  of  the  above  types  has  been  taught,  the  work  of  sections 
158,  168,  180  of  text  may  be  used  as  review  work. 

24.  Language  Forms  in  Division  of  Fractions: 

The  following  are  the  language  forms  to  be  used  in  the  work  of  the 
division  of  fractions : 

a.  Divide  f  by  |. 

Z).  How  many  times  is  1§  contained  in  20? 

Pupils  often  fail  upon  this  type  because  they  do  not  realize  which 
number  is  the  dividend  and  which  is  the  divisor.  First  teach  the 
principle  of  the  work  by  using  integers. 

25.  Given  Part  to  Find  Whole: 

Perhaps  the  most  difficult  of  all  problems  in  fractions  is  the  following : 
18  is  f  of  what  number?  (Text  pages  59,  139,  140,  141,  142,  143,  144.) 
The  solution  should  be  as  follows: 
f  of  the  number  =  18. 
I  of  the  number  =6. 
The  number  is  24. 
Pupils  should  be  able  to  show  the  meaning  of  the  above  statement  by 
means  of  a  diagram,  also  be  able  to  tell  why  18  is  divided  by  3,  not  4, 
to  find  \  of  the  unknown  number.     Show  pupils  why  the  following  form 
of  work  is  incorrect :     f  =  18,  etc. 

(10) 


&.  i--9^?4 

8      1 
d2|-.-6=3X^  = 

4 
'9 

/■    J-  ^  24  =  '?       ^ 

/i.  4a  -^  6f  =  ? 

b.  Simplify  :     ^— — -      Pupils  should  be  first  taught  to  read   ~~  as  137^ 

divided  by  200.      As  a  second  step  have  pupiLs  write  the  problem 
thus :  1371  -f-  200  =  ? 

c.  What  fraction  of  f  is  -J?     (Text  pages  131,  145,  146.) 
f  is  what  fraction  of  ^  ? 

22  is  what  fraction  of  22  ? 

d.  What  is  the  ratio  of  7  to  15  ?     Of  |  to  f  ?     Of  16  to  4 ?     (Text  pages 

54,  55.  56,  91-92.     Pages  130,  19-20.     Pages  145-146.) 

QUESTIONS  ON  FRACTIONS. 

26.  The  following  questions  and  the  tests  of  the  next  section  will  show 
in  a  general  way  the  knowledge  of  fractions  wiiich  a  child  should 
have  upon  completing  the  w^ork  of  this  topic.  These  questions 
will  also  serve  as  a  guide  for  Normal  students  desiring  to  prepare 
themselves  for  a  test  in  the  subject  of  fractions.  Questions  marked 
wdth  a  star  are  for  teachers  only. 

1.  Illustrate  two  different  types  of  improper  fractions. 

2.  Show  with  aid  of  a  diagram  that  |  or  |  of  a  line  is  the  same  as 

"I  of  the  line. 

3.  May  J  or  |  be  changed  to  12ths  ?     Give  a  reason  for  your  answer. 

4.  Illustrate  with  diagrams  what  is  meant  by  each  of  the  following 

statements : 
a.  Find  f  of  $12.     h.  $12  is  §  of  what  amount?     c.  i  is  what 
fraction  of  4?     d.  Find  f  of  f  of  a  line. 

5.  Which  of  the  following  fractions  may  be  reduced  to  lower  terms : 

i,  rs,  i>  ^2?     How  can  you  tell? 

6.  When  2  is  divided  by  f,  why  is  the  answer  greater  than  2? 
*7.  In  changing  -y-  to  a  mixed  number,  why  is  27  divided  by  5? 
*8.  In  changing  3f  to  an  improper  fraction,  why  is  the  whole  number, 

3,  first  multiplied  by  the  denominator  of  the  fraction,  f  ? 
*9.  Illustrate  the   difference  between  a  prime  number  and  an  odd 
number. 

10.  Why  must  unlike  fractions  be  changed  to  fractions  of  the  same 

denominator  before  they  may  be  added? 

11.  Read  Y-  in  two  ways. 

12.  Which  of  the  following  are  in  a  correct  form  to  be  cancelled? 

Change  those  that  are  not,  so  that  they  may  be  cancelled, 
a.  24X1=  ?  d  f  Xi=? 

h.  22.Xt=?  e.  f  X2i=? 

c.  8f  X9i=?  /.  f  X14=? 

13.  How  many  thirds,  fourths,  sixths,  are  needed  to  make  a  whole? 

14.  $18  is  f  of  what  amount?     Show  with  the  aid  of  a  diagram  why 

$18  is  divided  by  3  to  find  i  of  the  unknown  amount. 

15.  State  and  solve  a  problem  of  your  own  in  which  it  is  necessary 

to  invert  one  of  the  fractions. 


(11) 


3i  = 

if 

2i  = 

A 

6|  = 

A 

12 

If 

16.  Tell   what   is   wrong   with   each   of  the   following   problems   and 
rewrite  each  problem  in  its  correct  form: 

a.  J).  c. 

6f  6| 

Q2  3 

^4  8 

j6i_        2f 

15  Ol 6  U2 

■'■"2T  ^^¥  ^8' 

17.  In  the  multiplication  of  fractions,  tell  in  what  directions  numbers 

may  be  cancelled.  Is  it  possible  to  cancel  horizontally?  For 
instance,  can  you  cancel  the  4  and  8  in  |  X  -i\  =  ?  Give  a 
reason  for  your  answer. 

18.  Write  two  mixed  numbers  and  change  them  to  improper  fractions. 

19.  Find  the  smallest  common  denominator  for  the  following  frac- 

tions and  show  all  of  your  work : 

//       2  1  ^  2 

7)      ^  3  1  6_  3 

20.  Correct  the  following  statement: 

42  —  4 

21.  In  the  fraction  ^2^  which  term  is  the  numerator  and  Avhich  term 

the  denominator  of  the  fraction?     What  does  each  term  tell? 

22.  Change  the  following  fractions  to  36ths :  -j^,  |,  |. 

23.  Does  the  value  of  a  fraction  increase  as  its  denominator  increases? 
Is  ^i  of  a  line  greater  or  less  than  -j%  of  a  line  ? 

SUMMARY  OF  THE  FORMAL  WORK  IN  FRACTIONS. 

27.  The  following  tests  in  the  formal  work  of  fractions  are  intended  for 
pupils  of  the  grammar  grades  who  have  completed  the  work  in 
fractions.  Inexj^erienced  teachers  will  find  these  tests  useful  as 
guides  when  planning  reviews.  The  tests  will  also  serve  to  indicate 
to  Normal  students  some  of  the  fundamental  facts  in  fractions  with 
which  they  must  be  familiar. 

Test  in  Addition  and  Subtraction  of  Fractions. 

1.  Reduce  to  lowest  terms :  ^f ,  ff^,  |H,  ^S\,  if • 

What  are  the  tests  as  to  whether  or  not  fractions  may  be  reduced 
by  dividing  by  2,  3,  5,  or  10? 

2.  Classify  each  of  the  following  as  proper  or  improper  fractions 

or  mixed  numbers :     f,  If,  |,  -V ,  f- 

3.  Write  two  proper  fractions  which  may  be  reduced  to  lower  terms. 

Reduce  your  fractions. 

4.  a.  Write  two  improper  fractions.     Change  them  to  mixed  numbers. 
h.  Write  two  mixed  numbers  and  change  them  to  improper  frac- 
tions. 

5.  Find  the  sum  of:    S,%-,  8f,  2^,  3i,  |,  2-,%^,  2\,  S^.      Show  all  of 

your  work  for  finding  the  common  denominator. 

6.  Show  by  a  diagram  that    f ,  |  and  |  of  a  line  are  equal. 

7.  Change  f,  /a  to  24ths. 


(12) 


8.  Find  the  difference  between  20f  and  108f 

9.  Subtract  12f  from  25. 

10.  What  must  be  added  to  206$  to  make  1000^? 

11.  206f  -  ISf . 

Test  in  Multiplication  and  Division  of  Fractions. 

1.  Find  f  of  20f. 

2.  What  is  i  of  t? 

3.  How  many  times  is  |  contained  in  2|? 

4.  Divide  f  by  If. 

5.  $2f  is  I  of  what  amount? 

6.  Divide  1  ft.  9  in.  by  7  inches. 

'•  300     ~  ■ 

8.  17  is  what  fraction  of  29? 

9.  What  part  of  |  is  |  ? 

10.  f  are  what  part  of  |  ? 

11.  What  is  the  ratio  of  7  to  11  ?    Of  12  to  4?    Of  t  to  4? 

12.  If  i  of  a  yard  of  satin  is  worth  $1.60,  what  is  the  price  of  satin  per 

yard  ? 

13.  What  fraction  of  $2.10  is  $1.05? 

14.  (/.  12f  X7f=?     &.  3i--24=?     c.  |-f-t=? 

HOW  TO   TEACH  THE  CRITICAL  DIFFICULTIES  OF  DECIMALS. 

This  summary  of  type  problems  in  decimals  and  the  suggestions  made 
here  in  regard  to  the  form  and  the  method  of  work  to  be  employed  in 
the  teaching  of  this  subject  may  be  used  in  connection  with  the  indicated 
sections  of  the  California  State  Series  Advanced  Arithmetic  or  any 
other  arithmetic  text. 

Normal  students  will  be  expected  to  be  familiar  with  the  difficulties, 
form  of  work  and  methods  of  teaching  suggested  in  the  following 
sections. 

28.  Reading  and  Writing  Decimals : 

1.  Teach  pupils  to  read  and  write  decimals  through  millionths'  place. 

2.  Be  careful  to  see  that  pupils  pronounce  and  spell  correctly  the 
names  of  the  decimal  places.  In  writing  the  word  hundredths,  pupils 
often  omit  the  second  d.  In  pronouncing  the  word  hundredths,  pupils 
often  neglect  to  sound  the  th.  Point  out  that  100  and  one  hundredth 
are  not  the  same.  The  fourth  and  fifth  decimal  places  are  "ten- 
thousandths  ;  hundred-thousandths, ' '  not  tens  of  thousandths  or  hun- 
dreds of  thousandths. 

3.  Teach  that  the  hyphen  between  the  ten  and  thousandths  in  the 
word  three  ten-thousandths,  makes  the  two  words  one.  3  ten-thou- 
sandths is  written  decimally  as  .0003.  The  name  of  the  decimal  place 
is  ten-thousandths.     The  problem  requires  us  to  write  three  of  them. 

When  there  is  no  hyphen  between  the  ten  and  thousandths,  they  are 
to  be  regarded  as  separate  words.     Thus,  ten  thousandths  would  be 

3—1.5484  (13) 


written  decimally  .010.     The  name  of  the  decimal  place  is  thousandths. 
The  problem  requires  us  to  write  ten  of  them. 

4.  In  reading  a  number  such  as  273.003,  there  is  only  one  and. 
This  is  the  decimal  point.  The  above  number  should  be  read  ''tw^o 
hundred  seventy-three  and  three  thousandths."  Have  pupils  read 
$20.85,  1.04,  etc. 

5.  Ask :  What  is  the  name  of  the  third  place  after  the  decimal  point  ? 
The  fourth  place?  The  second  place,  etc.  How  many  decimal  places 
are  needed  to  express  ten-thousandths?     Hundredths?  etc. 

6.  When  w^riting  decimals  from  dictation,  insist  upon  pupils  placing 
the  decimal  point  as  soon  as  the  "and"  is  mentioned.  This  is  the  only 
way  in  which  pupils  will  acquire  the  habit  of  always  inserting  the 
decimal  point. 

7.  Teach  which  is  of  the  greater  value  and  why — 3  tenths.  3  hun- 
dredths or  3  thousandths,  etc.  Pupils  often  think  that  three  hundredths 
or  three  thousandths  of  a  quantity  is  greater  than  three  tenths  of  the 
same  quantity. 

8.  In  a  number  such  as  2.2222.  have  a  pupil  tell  the  value  of  each  2 
as  compared  with  the  2  to  its  right  or  left. 

9.  Show  pupils  that  .4,  .40,  .400  of  a  quantity  are  all  of  equal  value. 
Point  out  the  value  of  the  naught  in  numbers  such  as  the  above. 

10.  Show  pupils  how  the  naughts  change  the  value  of  each  of  the 
following  numbers :     .002,  .02,  1.02,  1.10. 

11.  Explain  that  our  system  of  money  is  a  decimal  system.  (Text, 
pages  9,  14.) 

12.  Have  pupils  see  that  .6  is  onlj*  another  way  of  writing  the  fraction 
1%.     Emphasize  the  fact  that  a  decimal  is  a  fraction  of  a  quantity. 

ADDITION  OF  DECIMALS. 

The  addition  of  decimals  is  merely  an  application  of  the  reading  and 
W'riting  of  decimals. 

Dictate  exercises  in  which  some  of  the  numbers  are  liuudredths, 
others  thousandths,  integers,  etc.  Do  not  permit  pupils  to  fill  out 
vacant  places  with  naughts.  Occasionally  write  the  decimals  out  in 
words  and  have  pupils  add  the  numbers. 

(For  drill  work,  see  Text  page  18.     Teach  also  the  Avork  of  page  14.) 

29.  Language  Forms : 

The  language  forms  to  be  used  in  the  addition  of  decimals  are: 

a.  Add  the  following :     .003,  .2,  1.46,  83,  .004. 

&.  Find  the  sum  of:     1.14,  89.2,  10006,  1.0012. 

c.  Work  in  the  addition  of  decimals  may  be  varied  by  using  the 
following  form : 

Write  in  the  order  of  their  value  and  add :  Sixteen ;  thirty-two  and 
four  hundred-thousandths ;  four  hundred-thousandths ;  one  and  three 
millionths;  eight  hundredths.  The  above  exercise  is  a  good  test  a.  of 
the  pupil's  ability  to  write  decimals;  h.  to  ascertain  their  relative  value. 

c.  Emphasize  the  fact  that  in  the  addition  of  decimals,  the  decimal 
points  must  be  kept  under  each  other. 

(14) 


SUBTRACTION  OF  DECIMALS. 

28.  Types  in  Subtraction  of  Decimals: 

a.  h.  c.  d.  e.  f. 

275.85         275.85         275.85  .7  295  296 

-14.25         -14.  -     .974        -.0024        -  14.006         -     .009 


Little  or  no  difficulty  will  be  experienced  in  teaching  the  subtraction 
of  decimals  provided  pupils  are  proficient  in  the  reading  and  writing 
of  decimals.  Emphasize  the  fact  that  decimal  points  must  be  kept 
under  each  other. 

29.  Language  Forms  for  Subtraction  of  Decimals. 

Often  pupils,  who  experience  little  difficulty  in  solving  subtraction 
exercises  in  which  the  minuend  and  subtrahend  are  correctly  placed, 
fail  when  confronted  with  a  problem  requiring  discrimination  between 
the  two  terms.  As  this  is  the  form  in  which  subtraction  is  usually 
encountered  in  life  and  even  in  the  problems  of  a  textbook,  it  is  neces- 
sary to  teach  the  various  language  forms  which  serve  to  indicate  that 
the  process  of  subtraction  is  to  be  employed.  Only  the  more  common 
forms  are  enumerated  below  in  the  order  of  their  difficulty : 

1.  Take  4.06  from  19.65. 

2.  Subtract  82.3  from. 963.64. 

3.  Find  the  difference  between  444.85  and  200. 

4.  Find  the  difference  between  .245  and  48.65. 

(This  last  type  is  especially  difficult  because  the  smaller  num- 
ber is  given  first.  Pupils  often  attempt  to  subtract  the  numbers 
in  the  order  in  which  they  appear  in  the  problem.) 

5.  What  must  be  added  to  8.56  to  make  27.2? 

6.  Teach  Text  page  25. 

7.  The  addition  and  subtraction  of  decimals  may  be  taught  in  a 
very  short  time  provided  pupils  can  read  and  write  decimals  with- 
out hesitation. 

MULTIPLICATION  OF  DECIMALS. 

30.  Two  Principles  to  Be  Taught : 

There  are  two  things  to  be  emphasized  in  teaching  the  multiplication 
of  decimals — (1)  the  correct  placing  of  the  decimal  point  in  the 
answer.  (2)  The  correct  placing  of  partial  products  with  the  different 
types  of  multipliers. 

31.  The  Placing  of  the  Decimal  Point: 

1.  Do  not  insist  upon  pupils  writing  the  multiplier  so  that  its  decimal 
point  is  under  the  decimal  point  of  the  multiplicand.  This  form  often 
causes  pupils  to  subtract  instead  of  multiplying  the  exercise.  Use  this 
form : 

2.4  2  9  6  ^^.  2.4  2  9  6 

X     .9  6  "'^  X.9  6 


do) 


2.  Do  not  use  multiplicands  or  multipliers  which  have  a  great  many 
figures.  The  fatigue  induced  by  the  working  of  such  long  problems 
leads  tf>  many  mechanical  errors.  Two  short  problems  are  better  than 
one  long  one  for  teaching  a  new  principle. 

TYPES  OF  MULTIPLIERS  AND  MULTIPLICANDS. 

The  tj'pe  problems  of  multiplication  are  enumerated  below  in  the 
order  of  their  difficulty. 

The  same  general  form  and  method  of  work  should  be  employed  in 
teaching  pupils  to  place  their  partial  products  when  using  each  and 
every  type  of  multiplier.  The  one  rule,  "Place  your  first  answer 
directly  under  the  number  you  have  multiplied  by"  gives  excellent 
results. 

32.  Types  of  Multipliers : 

a.  b.  c.  d.  e.  f. 

24.82  900.2  427.08  6.347  14.2  .007 

X.422         X  2.5         X 6.009  X  4.80         X-06|         X  .003 


1.  In  type  a,  the  emphasis  is  only  upon  the  correct  placing  of  the 
decimal  point  in  the  answer. 

2.  In  type  &,  the  naughts  of  the  multiplicand  offer  a  difficulty  to 
many  pupils. 

3.  In  types  c  and  d,  pupils  often  fail  to  place  the  partial  products  in 
their  correct  place.  Use  the  suggestion  of  "Place  your  first  answer 
directly  under  the  number  you  have  multiplied  by"  and  the  difficulty 
wall  disappear. 

4.  In  type  e,  be  careful  to  see  that  pupils  do  not  consider  the  fraction 
as  occupying  a  decimal  place  and  also  that  the  product  obtained  by 
multiplying  by  6  is  correctly  placed. 

5.  In  type  f,  teach  pupils  that  naughts  must  be  added  in  the  product 
to  the  left  of  the  integer.  Show  why  naughts  must  be  added  to  the  left 
and  not  to  the  right  of  the  integer.  Pupils  often  subtract  instead  of 
multiplying  this  tj^pe  of  problem  as  its  appearance  is  very  much  that  of 
a  subtraction  exercise. 

33.  Language  Forms  To  Be  Used  In  Connection  With  the  Multiplica- 
tion of  Decimals: 

1.  Find  the  product  of  24.84  and  .32. 

2.  What  is  six  hundredths  of  24.84? 

3.  Find  decimally  /v  of  iVoV- 

4.  Multiply  .002  by  .003. 

DIVISION  OF  DECIMALS. 

In  teaching  the  division  of  decimals,  follow  closely  each  of  the  fol- 
lowing steps : 

1.  Be  certain  that  pupils  know  the  difference  between  the  terms 
dividend  and  divisor. 


(16) 


2.  Write  on  the  board  division  problems  in  which  there  are  more 
decimal  places  in  the  dividend  than  in  the  divisor.  Write  other 
problems  in  which  there  are  less  decimal  places  in  the  dividend  than 
in  the  divisor. 

3.  Teach  that  there  must  always  be  at  least  as  manj-  decimal  places 
in  the  dividend  as  in  the  divisor.  Emphasize  the  fact  that  the  dividend 
may  have  more  decimal  places  than  the  divisor. 

From  the  problems  on  the  board,  have  pupils  select  those  in  which 
the  relation  between  the  number  of  decimal  places  in  the  dividend  and 
the  divisor  is  correct. 

4.  Teach  pupils  that  naughts  are  added  to  dividends  that  need  more 
decimal  places.  Show  why  these  naughts  do  not  change  tlie  value  of 
the  decimal. 

5.  Dictate  many  problems  to  pupils  and  have  them  fix  the  dividends 
that  need  attention.  Do  not  leave  this  step  of  the  w^ork  until  it  is 
thoroughly  mastered. 

6.  Placing  Decimal  Point  in  Answer : 

Use  the  following  method  to  teach  pupils  to  place  decimal  points  in 
the  answer.  Insist  upon  pupils  placing  the  deciiual  point  in  the  answer 
before  the  problem  is  solved. 

a.  Write  the  following  problem  on  the  board : 


.0  0  2  )  4  1  8.2  8  4  9 
Place  your  chalk  on  the  decimal  point  iu  divisor  and  count  the 
number  of  decimal  places,  saying  "decimal  point— one,  two, 
three."  Place  your  chalk  on  the  decimal  point  in  the  dividend 
and  count  off  the  same  number  of  decimal  places,  saying 
"decimal  point — one,  two,  three."  Place  decimal  point  in 
quotient  above  and  after  the  last  number  you  counted.  Try 
other  types  and  iLse  the  same  language  form. 


.4  )   8  2.4  2  9 

c.  Dictate   problem   after  problem   and   have   pupils   count   off  the 

decimal  places  and  fix  the  decimal  point  in  the  cpiotient.  At 
this  stage  of  the  work,  do  not  permit  pupils  to  solve  the 
problems.  All  emphasis  should  be  upon  the  correct  placing  of 
the  decimal  points. 

d.  The  following  are  the  advantages  of  the  above  method :      (l)It    is 

not  necessary  for  pupils  to  memorize  any  rule  for  pointing  off 
the  answer  in  division  of  decimals.  Memorized  rules  are  soon 
forgotten.  (2)  The  decimal  is  placed  before  beginning  any  of 
the  work  of  division;  (3)  when  the  decimal  point  is  placed  in 
the  answer,  pupils  can  tell  at  a  glance  the  number  of  places  to 
which  the  answer  has  been  carried. 
34.  Carrying-  Answers  Out  to  Three  Decimal  Places : 

1.  When  pupils  can  correctly  place  the  decimal  point  in  ;uiy  type 
problem,  show  them  that  the  problem  is  really  one  in  the  division  of 
integers. 

(IT) 


2.  Teaeli  pupils  that  to  carry  an  answer  out  three  places  decimally, 
means  to  have  three  figures  in  the  answer  after  the  decimal  point. 
Pupils  sometimes  think  that  adding  three  naughts  to  the  dividend  has 
carried  the  answer  out  decimally  three  places. 

3.  After  carrying  the  answer  out  to  three  decimal  places,  a  plus  sign 
should  he  used,  not  a  common  fraction  remainder. 

Suggestions : 

1.  Insist  upon  pupils  alwaj^s  placing  the  decimal  point  in  the  answer 
before  attempting  to  perform  the  division  of  the  problem.  This  is  the 
onl}^  way  in  which  pupils  will  ever  acquire  the  habit  of  always  placing 
the  decimal  point  in  the  quotient. 

2.  Have  all  answers  carried  out  to  three  decimal  places. 

3.  Do  not  place  the  emphasis  upon  the  length  of  the  problem,  but 
upon  the  correct  placing  of  the  decimal  point. 

4.  Occasionally  make  use  of  the  divisor  3.1416  which  will  be  used 
later  in  the  work  of  mensuration. 

5.  Use  the  following  language  forms  when  dictating  problems: 

35.  Language  Forms  for  Division  of  Decimals. 

a.  How  many  times  are  .2  contained  in  42.006  ? 
h.  Divide  4.09  by  .6. 

c.  Divide  2  by  11. 

d.  $20.45  is  .006  of  what  amount? 

36.  Types  to  be  Found  in  the  Division  of  Decimals. 


.004 


9- 

Divide  2  by  7. 

i. 


6.5 

)   .0042 

.07 

d. 

)   63.2 

/. 

27 

)   1.0246 
h. 

Chi 

mge  I'l'  to  a  decimal. 

J- 

3.1416  )~22' 

$1.05  )   $728.45 

For  drill  work  after  all  the  above  principles  have  been  taught  see 
Text,  pages  68-69-70. 

FRACTIONAL  EQUIVALENTS. 

Have  the  pupils  memorize  the  following : 

.50  =i  .40  =f  .m  =  i 

.33i  =  i  .60    =f  .37i  =  f 

.66f  =  |  .80    =f  .62|  =  f 

.25    =i  •                .161  =  i  .S7i  =  x 

.75    =1  .83^  =  1  .10    =Jo 

.20   =i  .141  =  ^  .30   =xV 
In  connection  with  this  section  teach  the  material  on  pages  156,  157 
and  158  of  text. 


(IS) 


37.  To  Divide  Integers  and  Decimals  by  10,  100,  1000. 

The  following  are  the  types  of  numbers  which  pupils  should  be  taught 
to  divide  first  by  10,  then  100,  then  1000  (text  pages  37,  40)  :  135,  40, 
400,  6,  .2,  .06,  1.3,  $4,  $200,  $1.60,  $.85,  $.40. 
a.  Write  4.6  on  the  board.     Erase  the  6.     This  leaves  4  with  a  decimal 

point  after  it.     Have  pupils  read  this  number. 
h.  Write  16,  18,  32  on  the  board.     Have  pupils  locate  the  decimal  point 
in  each  of  these  numbers. 

c.  Teach  that  to  divide  16,  18,  32,  etc.,  by  10,  the  decimal  point  is  moved 

one  place  to  the  left. 

d.  Teach  that  numbers,  such  as  50,  60,  40,  etc.,  are  divided  by  10  by 

moving  the  decimal  point  one  place  to  the  left.  Emphasize  the  fact 
that  such  numbers  are  divided  by  10,  not  by  crossing  off  the 
naughts  but  by  moving  the  decimal  point  one  place  to  the  left. 

c.  Care  must  be  taken  to  see  that  pupils  do  not  make  mistakes  in 
dividing  numbers  such  as  $.40,  $1.40,  etc.,  by  10.  Ask:  ''Does 
crossing  off  the  naught  divide  the  above  numbers  by  10?"  "How 
are  they  divided  by  10  ? " 

/.  Ask:  "How  is  a  number  divided  by  10?" 

g.  When  the  decimal  point  has  been  moved  one  place  to  the  left,  what 
has  been  done  to  the  number  ? 

h.  Repeat  the  same  steps  in  teaching  pupils  to  divide  numbers  by  100 
and  1000.     Use  the  types  listed  above. 

(.  Ask:  "How  are  numbers  divided  by  10,  100,  1000?"  When  deci- 
mal points  are  moved  one,  two  or  three  places  to  the  left,  what  has 
been  done  to  the  number  ? 

38.  To  Multiply  Numbers  by  10,  100,  1000. 

a.  Teach  that  numbers  are  multiplied  by  10,  100,  1000  by  moving  the 

decimal  point  one,  two,  or  three  places  to  the  right. 
h.  Ask:  "How  is  a  number  multiplied  by  10?     100?     1000?" 

c.  When  is  the  decimal  point  moved  to  the  right  ? 

d.  When  is  the  decimal  point  moved  to  the  left? 

e.  Put  a  column  of  nuinbers  on  the  board.     Mix  in  decimals  with  the 

integers.  Call  upon  pupils  to  give  the  answers  when  the  various 
numbers  are  divided  by  10,  100,  1000,  or  when  multiplied  bv  10. 
100,  1000. 

MISCELLANEOUS  QUESTIONS  ON  DECIMALS. 

After  all  of  the  principles  in  decimals  have  been  taught,  the  following 
questions  and  the  tests  of  the  next  section  may  be  used  to  ascertain  the 
extent  of  the  pupils'  knowledge. 

Normal  students  will  find  these  c|uestions  and  tests  useful  in  tlieir 
preparatory  work  in  arithmetic. 

1.  Write  in  their  order  the  names  of  the  decimal  places  through 
millionths. 

2.  Explain  why  our  system  of  money  is  a  decimal  system. 

3.  Which  is  of  the  greater  value,  $1.10  or  1100  mills?  How  much 
greater  ? 

(10) 


4.  Write  in  figures  three  hundred  thousandths;  three  hundred- 
thousandths  ;  three  hundred  thousand.  How  do  you  distinguish  each 
type? 

5.  In  the  expression  111.111,  how  does  each  1  compare  in  value  with 
the  one  to  its  left  ? 

6.  If  a  naught  is  added  to  the  right  of  $2.10.  has  the  number  been 
multiplied  by  ten?  If  the  naught  at  the  end  of  $2.10  is  crossed  out, 
has  the  number  been  divided  by  ten  ?     Explain  in  full. 

7.  What  is  the  difference  in  value  between  100  and  .01  ? 

8.  Where  4s  the  decimal  point  in  a  number  such  as  24  ? 

9.  If  the  decimal  .9  is  a  fraction,  where  is  its  numerator  and 
denominator  ? 

10.  In  a  decimal  expression  such  as  .665,  does  the  fraction,  f,  count 
as  a  decinial  place  ? 

11.  Show  that  .8,  .80,  .800  of  a  quantity  are  all  of  equal  value. 

12.  Write  the  table  of  United  States  money. 

13.  In  the  multiplication  of  decimals,  how  do  you  know  the  number 
of  decimal  places  to  point  off  in  the  answer  ? 

TEST  IN  ADDITION  AND  SUBTRACTION  OF  DECIMALS. 

1.  Express  each  of  the  following  decimals  as  common  fractions :  .024, 
.004,  .8,  2.101,  .400,  .00265. 

2.  If  two  naughts  are  added  to  the  right  of  the  1  in  the  decimal 
expression  .1,  what  is  the  effect  upon  the  value  of  the  decimal?  If  two 
naughts  are  added  to  the  left  of  the  1  in  what  way  is  the  value  of  the 
decimal  .1  changed? 

3.  How  does  .4  compare  in  value  with  .44  and  with  .004  ? 

4.  Show  that  .1,  .10,  .100  are  of  equal  value. 

5.  What  is  the  name  of  the  fourth  decimal  place?  The  third  deci- 
mal place  ?     The  second  decimal  place  ? 

6.  Write  at  least  five  aliquot  parts  of  a  dollar. 

7.  Arrange  the  following  decimals  in  the  order  of  their  value  and  add : 
Six  hundred  six  and  six  hundred-thousandths ;  eighteen  and  one  mill- 
ionth ;  twelve  and  six  tenths ;  five  hundred  seven  thousandths ;  six  hun- 
dred; fifteen  hundredths. 

8.  Find  the  difference  between  .1  and  1.426. 

9.  a.  h. 
296  A                              $110 

^  18.008  —  $       .85 


10.  Find  out  by  changing  to  common  fractions  whether  you  would 
rather  have  .5  of  a  pie.  .50  of  a  pie,  or  .500  of  the  same  pie. 

11.  What  is  the  difference  in  value  between  (a)  1  and  .1;  (h)  1.  and 
.01;  (c)  1.  and  .001;  {d)  100  and  .01  ? 

12.  Place  the  decimal  point  in  each  of  the  following  numl^ers :  sixteen 
hundredths;  eighty-five;  five  thcusandths;  two  hundred  seventy-five; 
sixteen  and  four  thousandths ;  twelve. 

13.  Write  decimally  and  add :  four  dollars  and  sixty-five  cents ; 
twenty-five    and    seven    tenths    dollars;    forty-eight    dollars    and    nine 

(20) 


cents;  eight  thousand  three  hundred  sixteen  dollars  and  sixteen  cents 
four  mills ;  four  and  sixteen  hundredths  dollars ;  eight  dollars  and  four 
cents ;  nine  mills ;  three  and  two  hundred  fifteen  thousandtlis  dollars. 

14.  What   is    .1   of   a   dollar?     One   hundredth   of   a   dollar?     One 
thousandth  of  a  dollar? 

TEST  IN  MULTIPLICATION  AND  DIVISION  OF  DECIMALS. 

1.  Find  the  product  of  80.04  and  40.8. 

2.  Divide  each  of  the  folloMing  problems  in  the  shortest  way  possible : 


$1.10  )   $240  300  )   150.1  400  )   .8394 

$600  )   $84.80  $.10  )  $1.10  .40  ^400 

3.  Change  -^f  to  a  decimal.     (Carry  quotient  to  three  decimal  places.) 

4.  Multiply  .004  by  .003. 

5.  Place  only  the  decimal  point  in  the  quotient  of  each  of  the  follow- 
ing problems : 

a.  3.1416  )r259         h.  .003  )   2.429         c.  .019  yii        d.  34  )   802.34 
e.  .7  y^  /.    19  Yl  g.  $.01  Y^       /'•  $1  )  $1.01 

6.  Divide  2  by  9. 

7.  ]Multiply  each  of  the  following  numbers  in  the  shortest  way  pos- 
sible, first  by  10.  then  100,  and  then  1000: 

$.45,  $1.40,  .0008,  .6,  .94,  $1,008,  2.75,  .009,  46.,  $.05. 

8.  Find  decimally  fo  of  42.094. 

9.  What  is  6|  hundredths  of  245.6  ? 

10.  Write  each  of  the  following  as  decimals :     f,  f,  |,  f ,  =.  |,  f,  ^,  f,  -|. 

RATIO,  AVERAGE,  TABLES,  BILLS  AND  RECEIPTS. 

The  following  questions  cover  several  miscellaneous  topics.  The 
questions  will  indicate  the  principles  to  be  taught  under  each  topic. 

39.  Test  on  Ratio  and  Average. 

1.  What  is  the  ratio  of  a  pound  to  an  ounce? 

2.  From  the  following  table,  find  Fred's  average  gain  in  weight  and 
in  height  each  year : 

Age.  Height.  Weight. 

11  yr.  4'  11"  98  lb. 

12  "  5'     2"  104  lb.  4  oz. 

13  "  5'     ^"  107  lb.  9  oz. 

14  "  5'     4"  110  lb.  1  oz. 
a.  Be  certain  that  pupils  first  find  the  gain  in  height. 

6.  Since  there  were  3  gains,  the  actual  gain  is  divided  by  3  to  find  the 
average  gain  in  height. 

3.  The  expenses  of  a  picnic  were  $3.40.  Divide  this  amount  between 
Mrs.  Brown  and  Mrs.  White  in  the  ratio  of  2  to  3. 

4.  ]\Iiss  Saunders  had  57  pupils  in  her  class.  There  were  42  pupils  in 
]\Iiss  Ryan's  class.  Find  the  ratio  of  the  number  of  pupils  in  Mi.ss 
Saunders'  class  to  the  number  in  Miss  Ryan's  class. 

5.  In  1890  the  population  of  San  Francisco  wa&  298,997.  In  1906 
the  population  of  San  Francisco  was  450,000.  Find  the  ratio  of  the 
population  in  1906  to  the  population  in  1890. 

(21) 


6.  During  the  month  of  September  Mrs.  Fisher  used  645  cubic  feet  of 
gas;  in  October,  750  cubic  feet;  in  November,  1,200  cubic  feet;  in 
December,  1,950  cubic  feet ;  in  January,  2,645  cubic  feet.  Find  the 
average  gain  in  the  number  of  cubic  feet  of  gas  used  each  month. 

7.  What  is  the  ratio  of  f  to  f  ? 

8.  Is  the  ratio  between  two  quantities  ever  expressed  as  a  concrete 
number  ?     Why  ? 

9.  Is  the  ratio  of  7  to  9  the  same  as  the  ratio  of  9  to  7 "? 

10.  During  a  certain  year  the  rainfall  for  six  months  in  San  Francisco 
was  as  follows :  December,  5.11  inches ;  January,  4.98  inches ;  Feb- 
ruary, 3.72  inches;  March,  3.28  inches;  April,  2.14  inches;  May,  0.68 
inches.     Find  the  average  rainfall  for  each  month. 

40.  Tables:     Linear,  Square,  Weights,  Country,  Time,  Liquid. 

a.  How  many  pints  of  milk  may  be  poured  into  a  five-gallon  can? 
h.  Mr.  Bray  owns  a  section  of  land.     What  is  the  value  of  the  entire 
farm  if  one  acre  is  worth  $75  f 

c.  What  is  the  value  of  956  eggs  at  $.32  a  dozen  ? 

d.  Helen  practices  two  and  one  half  hours  a  daj^  on  the  piano.  How 
many  minutes  does  she  practice  each  day? 

e.  At  $11.50  per  ton,  what  is  the  value  of  16,480  pounds  of  hay? 

/.  I  bought  2|  pounds  of  sugar.  How  many  ounces  of  sugar  did  I 
purchase  ? 

g.  What  is  the  cost  of  a  great  gross  of  pencils  if  one  pencil  costs  l^ff. 

h.  ]Mr.  Brown  owns  property  having  a  frontage  of  |  of  a  mile. 
Express  the  width  of  the  lot  in  feet. 

i.   A  square  foot  is  how  many  times  as  great  as  a  square  inch ! 

j.  Which  is  the  greater  in  value,  $2.10  or  8,124  mills?  How  much 
greater  ? 

A'.  How  many  steps,  each  12  inches  long  will  a  man  take  in  walking 
one  mile  ? 

/.  What  is  the  length  and  wndth  in  feet  of  a  square  room,  the  floor  of 
which  has  an  area  of  one  square  rod? 

m.  Find  the  sum  of  4'  6" ;  -}"  ■  V  i"  ■  2'  IV'. 

n.  Find  the  difference  between  102^'  and  ^". 

0.  How  many  dozen  are  there  in  a  gross  ?  Ounces  in  a  pound  ? 
Pounds  in  a  hundredweight?  Minutes  in  an  hour?  Units  in  a  score? 
Quarts  in  a  gallon?  Pints  in  a  quart?  Mills  in  a  cent?  Pounds  in  a 
short  ton  ?     Hundredweight  in  a  long  ton  ? 

II. 

a.  If  a  dealer  has  288  pencils  in  stock,  are  there  enough  to  fill  an  order 
for  a  gross  of  pencils? 

b.  Which  is  the  larger  amount,  18  cents  or  25  mills? 

c.  How  many  pounds  are  there  in  a  long  ton?  How  many  cwt.  are 
there  in  3485  pounds? 

d.  Alice  practiced  85  minutes  on  the  piano.  How  much  more  or  less 
than  an  hour  did  Alice  practice? 

e.  How  many  pint  bottles  may  be  filled  from  a  gallon  of  milk? 

(22) 


/.  Last  Christmas  I  sent  to  New  York  a  package  weighing  22  ounces. 
How  much  more  or  less  than  a  pound  did  the  package  weigh? 

g.  How  many  cubic  inches  are  there  in  a  gallon? 

h.  How  many  cords  of  wood  are  there  in  15,744  cubic  feet  of  lumber? 

i.    How  many  pounds  of  coal  are  there  in  2  hundredweight? 

j.   How  many  minutes  are  there  in  2  hours  45  minutes? 

A'.  How  many  years  are  there  in  "four  score  and  seven  years"? 

1.   How  many  hundredweight  are  there  in  one  ton? 

m.  If  a  ton  of  coal  is  worth  $14,  what  is  the  value  of  three  hundred- 
weight of  coal? 

n.  A  gallon  of  water  is  the  same  as  how  manj'  cubic  inches  of  water? 

41.  Bills  and  Receipts.     (Text  pages  42-44.) 

1.  Mr.  Berryman  rented  a  house  at  1207  Sacramento  Street  from  Mr. 
J.  R.  Hansen  for  the  sum  of  $35  a  month.  On  May  1st,  Mr.  Berryman 
paid  the  rent  for  one  month  in  advance.  Write  the  receipt  Mr.  Hansen 
gave  to  ]Mr.  Berryman. 

2.  ]\Iake  out  and  receipt  the  following  bill :  I\Irs.  J.  H.  Moore,  resid- 
ing at  3746  Sacramento  Street,  bought  of  her  dealer  the  following: 
January  3,  1914,  4  yd.  blue  ribbon  @  $.30;  2^  yd.  lace  @  $.75;  6 
handkerchiefs  @  $.25;  |  yd.  lace  yoking  @  $2.25. 

3.  Make  out  a  receipt  of  your  own  showing  that  money  has  been  paid 
by  you  to  some  other  person  for  work  done. 

4.  Make  out  and  receipt  the  following  bill  of  goods  purchased  from 
your  grocer  by  you  on  May  1st: 

3  lb.  coffee @  $.35 

1  tb.  tea @     .80 

2  doz.  eggs @     .30 

1  roll  of  butter__@     .65 

5.  Mrs.  Adams  bought  from  ]\Irs.  Dean  a  sewing  machine  for  the  sum 
of  $60.  Write  a  receipt  showing  that  Mrs.  Dean  has  received  the  money 
from  Mrs.  Adams. 

AREA  OF  RECTANGLES. 

(Text  pages  80-84.) 
The  material  of  pages  80,  81,  82,  83  of  the  Advanced  Arithmetic  may 
be  supplemented  with  much  additional  work  from  Bulletin  No.  31,  pages 
1-26. 

42.  Test  on  Area  of  Rectangles.     (Text  pages  80-84.) 

The  following  questions  may  be  used  to  test  a  pupil  upon  the  work 
of  pages  80  to  84  of  text. 

1.  Draw  three  different  figures,  each  of  which  is  a  rectangle.     Label 

the  base  and  altitude  of  each  figure  and  shade  its  surface. 

2.  Illustrate  with  diagrams  the  difference  between  a  horizontal  rect- 

angle and  a  square. 

3.  Illustrate  with  diagrams  the  difference  between  two  square  inches, 

two  inches  and  a  two-inch  square. 

4.  Draw  on  your  paper  four  right  angles,  each  of  which  is  in  a 

different  position. 

(23) 


5.  Is  a  very  thin  sheet  of  tissue  paper  a  surface  or  a  solid  ?     Give  a 

reason  for  your  answer. 

6.  Illustrate  by  means  of  a  diagram  which  is  the  larger,  a  square 

mile  or  a  section  of  land. 

7.  A  lot  is  125  feet  deep  and  has  a  frontage  of  25  feet.     What  is 

the  perimeter  of  the  lot? 

8.  How  many  feet  of  picture  molding  are  needed  for  a  room  21  feet 

long  and  16  feet  wide? 

9.  Make  a  list  of  three  rectangular  surfaces  that  appear  in  your 

classroom. 

10.  Find  the  perimeter  and  the  surface  of  this  page  of  your  book. 

11.  A  forest  extends  over  two  sections  of  land.     How  many  acres  of 

land  are  covered  b}'  the  forest  ? 

12.  What  is  the  length  in  feet  of  the  side  of  a  square  field,  the  area 

of  which  is  one  acre? 

13.  Draw  two  oblique,  two  horizontal  and  two  vertical  parallel  lines. 

14.  Draw  a  sc^uare  foot  on  your  paper.     (Let  a  line  6  inches  long 

represent  a  foot.)     Divide  your  square  foot  into  the  greatest 
possible  number  of  square  inches.     What  is  the  answer  ? 

15.  Draw  a  square  yard  on  your  paper.     (Let  6  inches  represent 

1  yard.)      Show"  by  your  diagram  the  number  of  square  feet 
there  are  in  one  square  yard. 

16.  What  is  the  perimeter  and  what  is  the  area  of  a  field  one  mile 

scpare?     Diagram  your  field. 

17.  Tell  for  what,  in  the  scpare  measure  or  the  linear  measure,  each 

of  the  following  numbers  stands :     640,  3,  12,  9,  320,  160,  30^, 
208f ,  16|,  144,  5i,  5,280. 

18.  With  a  diagram,  show  the  number  of  square  inches  there  are  on 

the  surface  of  a  three-inch  sc|uare.     What  is  its  area? 

19.  Draw,  if  you  can,  a  rectangle  which  contains  no  right  angles. 

20.  Illustrate  with  marked  diagrams  the  difference  between  the  perim- 

eter and  the  surface  of  a  four-inch  square. 

21.  Express  14|  square  feet  as  scjuare  inches.     Change  4  acres  to 

square  rods. 

22.  Draw  a  horizontal  rectangle,  a  vertical  rectangle  and  a  square. 

Label  the  base  and  altitude,  and  shade  the  surface  of  each 
figure. 

23.  Draw  a  line  which  is  perpendicular  to  another  line. 

24.  If  the  area  of  a  rectangle  is  given  in  acres  and  the  length  of  one 

side  in  rods,  how  is  the  perimeter  of  the  rectangle  found? 
Illustrate  with  a  problem  of  your  own. 

25.  If  the  area  of  the  floor  of  a  square  room  is  one  square  rod.  what 

is  the  length  and  width  of  the  room  expressed  in  feet? 

26.  At  $175  an  acre,  what  is  the  value  of  a  field  which  is  400  rods 

long  and  ^  of  a  mile  wide? 

27.  How  many  rails,  each  16|  feet  long,  are  needed  to  enclose  a  square 

field,  the  area  of  which  is  one  sc^uare  mile? 


(20 


28.  A  field  contains  3  acres  of  land  and  is  ^  of  a  mile  long.     What  is 

the  Avidth  of  the  field? 

29.  If  the  area  of  a  field  is  10  acres  and  one  side  is  20  rods  long, 

what  is  the  perimeter  of  the  field? 

30.  ]May  an  acre  or  any  part  of  an  acre  ever  be  used  to  measure  the 

length  of  a  line  ?     Give  a  reason  for  your  answer. 

31.  j\Ir.  Jones  paid  $550  for  his  farm.     Each  acre  cost  him  $55.     If 

the  length  of  the  farm  is  180  rods,  what  is  its  perimeter  ? 

CUBIC  MEASURE. 

(Text  pages  84-86.) 
The  material  of  pages  84,  85,  86  of  the  text  may  be  supplemented  with 
much  additional  work  from  pages  55  through  68  of  Bulletin  No.  32. 

43.  Test  on  Cubic  Measure.     (Text  pages  84^86.) 

The  following  questions  will  serve  as  a  test  of  a  pupil's  knowledge 
after  he  has  completed  pages  84  through  86  of  text. 

1.  How  many  cords  of  wood  are  there  in  a  pile  which  is  12  feet  long, 

8  feet  wide  and  4  feet  high? 

2.  How  many  cubic  yards  of  dirt  were  removed  from  a  lot  114'  X  50' 

if  the  earth  was  removed  to  a  depth  of  6  feet  ? 

3.  A  rectangular  tin  can  is  8"  X  10"  and  16  inches  high.     How 

many  gallons  of  water  will  the  can  hold? 

4.  A  swimming  tank  is  110  feet  long,  75  feet  wide  and  has  a  depth 

of  5  feet.  How  many  gallons  of  water  are  there  in  the  tank 
when  it  is  one  half  full?     (7^  gal.  =  1  cu.  ft.) 

5.  Illustrate  with  a  drawing  the  difference  between  a  two-inch  square 

and  a  two-inch  cube. 

6.  How  many  faces  has  a  cube  ?     "What  is  the  volume  of  a  three-inch 

cube? 

7.  Is  a  sheet  of  paper  a  surface  or  a  solid  ?     Why  ? 

8.  In  order  to  find  the  volume  of  a  prism,  how^  many  dimensions  must 

be  known?  State  and  solve  a  problem  of  your  own  in  which 
you  are  required  to  find  the  volume  of  a  prism. 

9.  A  gallon  of  water  is  the  same  as  how  many  cubic  inches  of  water? 

10.  If  the  length  and  width  of  a  room  are  given,  tell  whether  or  not 

you  can  find  (a)  the  area  of  the  ceiling,  (6)  the  volume  of  air 
in  the  room,  or  (c)  the  number  of  feet  of  picture  molding 
needed  for  the  room. 

11.  ]\Ir.  Harris  removed  one  foot  of  dirt  from  a  lot  having  a  depth  of 

114  feet  and  a  frontage  of  75  feet.  How  many  cubic  yards  of 
earth  were  removed  from  the  lot  ? 

12.  What  is  the  volume  of  a  four- inch  cube  ? 

13.  A  rectangular  vessel  is  26  inches  long,  18  inches  wide  and  9  inches 

deep.     What  is  its  capacity  expressed  in  gallons? 

14.  Write  the  table  of  cubic  measure. 


(2.5) 


PERCENTAGE. 

(Text  pages  166-190.) 

The  work  of  each  of  the  three  cases  in  percentage  may  be  divided  into 
two  parts — formal  work  and  problem  work.  Bulletin  No.  29  presents. 
in  the  order  of  their  difficulty,  each  of  the  points  to  be  taught  under 
the  formal  and  problem  work  of  each  of  the  cases  in  percentage.  As 
this  bulletin  is  for  self -instruction,  it  contains  all  necessary  explanations 
of  difficulties.  Normal  students  are  expected  to  be  familiar  with  all 
of  the  problems  of  this  bulletin  as  well  as  the  problems  in  the  State  Text 
on  pages  166-190. 

The  mechanical  or  formal  work  of  each  case  in  percentage  should  be 
mastered  before  the  problem  work  is  introduced.  For  this  reason,  the 
formal  work  of  each  case  of  percentage  is  taught  before  pupils  are 
required  to  center  their  attention  upon  the  problem  work  of  a  given 
case.  Most  of  the  complicated  and  impractical  problems  which  fall 
under  the  second  case  in  percentage  have  been  omitted  from  Bulletin  29. 

43.  Formal  Work  of  Case  One  in  Percentage. 

a.  The  most  difficult  piece  of  formal  Avork  in  the  first  case  in  per- 
centage is  the  changing  of  the  various  types  of  per  cents  to  correct 
decimals. 

&.  Pupils  should  be  taught  to  change  each  type  of  per  cent  as  a 
separate  step  of  work.  As  soon  as  possible,  work  should  be  given  which 
will  be  an  application  of  the  type  taught.  Unless  this  is  done,  pupils 
will  think  of  the  changing  of  per  cents  to  decimals  as  a  line  of  work 
separate  from  the  rest  of  percentage. 

c.  Bulletin  No.  29,  pages  3  through  23,  presents  in  the  order  of  their 
difficulty,  each  of  the  difficulties  that  occur  in  the  formal  work  of 
Case  One. 

d.  The  following  are  the  most  difficult  sections:  7,  8,  11,  12,  16,  IS, 
22,  25,  26,  28,  29,  30.  Teachers  should  question  pupils  closely  upon 
each  of  these  sections  before  permitting  them  to  take  up  any  new  work. 

e.  The  tests  of  sections  32  and  33  may  be  used  as  guides  Avhen  making 
out  tests  upon  the  formal  work  of  Case  One. 

Problem  Work  of  Case  One  in  Percentage. 
The  failure  of  many  pupils  in  the  problem  work  of  arithmetic  may 
be  traced  (1)  to  their  inability  to  read  a  problem  and  ascertain  what 
is  given  in  the  problem  and  what  is  recpiired  to  be  found;  (2)  to  lack 
of  ability  to  understand  the  situation,  words  and  phrases  of  the  prob- 
lem; (3)  helplessness  in  attempting  the  solution  of  the  problem;  (4) 
failure  to  test  the  answers  to  problems. 

44.  Statement  of  Problems : 

Pupils  should  be  required  to  write  a  statement  of  every  problem 
before  attempting  its  solution. 

When  a  pupil  can  write  a  correct  statement  of  what  is  given  in 
a  problem  and  what  is  to  be  found,  it  is  an  indication  to  the  teacher 
that  the  pupil  has  at  least  read  the  problem.     The  method  of  teaching 

(26) 


pupils  to  write  the  statement  of  a  problem  is  to  be  found  on  page  28  of 
Bnlletin  No.  29.  Pnpils  should  master  the  writing  of  statements  before 
being  permitted  to  proceed  to  the  next  step  of  work. 

45.  Wording  of  Problems: 

As  far  as  possible,  the  situations  in  the  problems  of  the  bulletin  are 
those  with  which  pupils  are  familiar  or  which  can  readily  be  explained 
to  them. 

46.  Solution  of  Problems  and  Proof : 

Under  each  of  the  tj'pe  problems,  questions  are  asked  to  help  pupils 
in  their  method  of  attacking  the  problem.  In  each  case  in  percentage, 
emphasis  is  placed  upon  having  the  pupil  first  w^ork  out  the  meaning 
of  the  per  cent  in  the  problem.  As  part  of  the  regular  problem  work, 
pupils  are  required  to  prove  the  answers  to  their  problems  whenever 
possible. 

The  problems  of  Case  One  on  pages  52  through  page  58  should  receive 
special  attention. 

The  tests  on  pages  60,  61  may  be  used  as  guides  when  planning 
tests  on  the  problem  work  of  Case  One  in  percentage. 

Formal  Work  Case  Two  in  Percentage. 

The  second  case  in  percentage  is  usually  more  difficult  for  pupils  to 
master  than  either  of  the  other  cases.  Experience  has  shown  that  pupils 
gain  more  power  is  attacking  problems  of  this  case  Avlien  taught  the 
method  of  unitary  analysis  rather  than  the  method  of  the  text.  All 
explanations  of  difficulties  of  the  formal  work  of  Case  II  of  percentage 
will  be  found  on  pages  62,  through  72  of  the  bulletin.  Care  must  be 
taken  to  see  that  pupils  follow  closely  the  work  of  the  models. 

The  suggestions  given  on  pages  69-70  are  important. 

The  tests  on  pages  74-75  may  be  used  as  guides,  when  planning  tests, 
on  the  formal  work  of  Case  II  of  percentage. 

The  Problem  Work  of  Case  Two  in  Percentage. 

Problems  similar  to  those  on  page  178  of  text  are  very  difficult  for 
pupils  to  understand  and  also  very  impractical.  Problems  of  this 
type  have  been  entirely  omitted  from  the  Avork  of  the  bulletin.  Normal 
students  are  however  required  to  be  familiar  with  such  problems. 

Review  of  Cases  One  and  Two. 

The  test  of  a  pupil's  knowledge  of  the  first  or  second  cases  in  per- 
centage comes  when  the  problems  of  these  two  cases  are  mixed.  Such 
a  review  may  be  found  on  pages  90-95  of  the  bulletin. 

Case  Three  Formal. 

Pupils  must  be  taught  to  change  decimals  to  per  cents  before  begin- 
ning the  real  work  of  Case  Three.  Each  type  of  decimal  is  taught  and 
applied  in  the  various  sections  of  pages  99  through  110  of  the  bulletin 
(Text  179-182).  A  test  of  the  formal  work  of  Case  Three  of  percentage 
may  be  found  on  page  107. 

(27) 


Problem  Work  of  Case  Three. 

The  difficulties  met  in  teaching  the  problems  of  Case  III  in  percentage 
are  explained  and  illustrated  on  pages  114  through  122  of  the  bulletin. 

Review  of  Cases  One,  Two,  Three. 

The  test  of  the  pupil's  knowledge  of  percentage  comes  when  problems 
of  the  three  eases  are  presented  in  a  miscellaneous  order.  Such  a  review 
may  be  found  on  pages  184  through  191  of  text  or  pages  122  through 
126  of  bulletin.  Normal  students  are  expected  to  be  familiar  with  all 
of  the  above  mentioned  problems. 

TEST  ON  FORMAL  WORK  OF  PERCENTAGE. 

The  following  miscellaneous  review  of  the  three  cases  of  percentage 
may  be  used  as  a  guide  when  it  is  necessary  to  test  a  pupil 's  knowledge 
of  the  three  cases  of  percentage.  At  the  beginning  of  a  term  or  when 
a  new  pupil  enters  a  class,  such  a  test  is  useful.  Pupils  can  be  then 
drilled  upon  the  points  in  which  their  knowledge  is  deficient. 

1.  Illustrate  with  a  diagram  and  a  problem  the  difference  between 

i  of  $120  and  ^%  of  $120. 

2.  Write  each  of  the  following  expressions  in  two  other  ways : 

lOlf  %,  f  %,  200%,  104%,  45%,  66§%,  f  %,  1^%,  1^  i%,  1^%, 
t%,  12%,  110%,  Jo,  liV%,  1%,  133^%,  8-^%,  250%,  6,  103%, 
82%,  101f%,  85%,  111%,  2. 

3.  $102  is  31%  of  what  amount? 

4.  Find  the  entire  cost  of  a  book  when  83-?,%  of  the  cost  is  $2.50. 

5.  AVrite  each  of  the  following  as  per  cents :     ^,  f ,  f ,  f ,  f ,  i,  ^,  fo, 

1    1 

6.  Change  ^|  to  a  per  cent. 

7.  What  per  cent  of  2  is  1.75  ? 

8.  What  per  cent  of  $810.20  is  $350  ? 

9.  Find  in  the  shortest  way— a.  166|%  of  270.     b.  8^%  less  than 

180. 

10.  Express  each  of  the  following  as  per  cents : 

a.  3.1,  .0401,  1.1,  2.3,  2|,  .0075,  .00125. 
h.  .8906,  .241,  .161,  1.001,  1.013,  2.5,  .024. 

c.  li  .0075,  .75,  .010,  1.011,  1.111,  3. 

d.  2i,  .022,  1.4,  1.067,  2|,  4.25,  3.2. 

11.  Find  the  number  that  is  17-|%  less  than  1,000. 

12.  $850  is  1|%  of  what  amount? 

13.  Find  100%  more  than  $600. 

14.  Find  Y/c  less  than  210. 

15.  AVhat  per  cent  of  a  mile  is  a  rod? 

16.  Which  is  the  largest  amount,  -jVir  1'^%  o^"  -^6  of  $250  ? 

17.  What  is  the  difference  between  130%  of  $210  and  30%^  more  than 

$210  ? 

18.  What  is  the  difference  between  5%  less  than  $115  and  95%  of 

$115? 

19.  Write  a  per  cent  of  a  quantity  that  is  equal  to  less  than  the 

whole  of  a  quantity.    To  more  than  the  whole  of  the  quantity  ? 
(28) 


20.  Why  is  it  incorrect  to  say  that  $400  =  125 7o  ? 

Correct  the  statement. 

21.  Tell  two  ways  of  finding  331%,  40%  or  25%  of  a  quantity. 

22.  Six  out  of  every  100  is  the  same  as  how  many  per  cent! 

23.  Is  there  any  difference  between  ^,  12^%o  and  i%  of  64?     Illus- 

trate with  work  and  a  diagram. 

24.  What  per  cent  of  I  is  f? 

25.  Tell  without  working  the  following  problems  in  which  cases  the 

answers  will  be  larger  and  in  which  cases  smaller  than  the  given 

cjuantity. 

a.  Find  105%o  of  650.     h.  What  is  4%o  less  than  220?     c.  Find 

i%  of  125.     d.  What  is  2^%  more  than  175?     e.  Find  200%o 

of  $150. 

26.  What  per  cent  of  a  quantity  is  always  equal  to  a  quantity  ? 

TEST  ON  PROBLEM  WORK  OF  PERCENTAGE. 

(Text  pages  171-190.) 
The  following  test  on  the  problem  work  of  the  three  cases  of  per- 
centage may  be  used  as  a  guide  when  planning  review  work  or  as  a 
test  for  new  pupils  who  enter  the  class  late  in  the  term.  Problems 
at  the  end  marked  with  stars  are  the  additional  types  with  which 
Normal  students  are  required  to  be  familiar. 

1.  Mr.  Hansen  paid  $6,000  for  a  piece  of  property.     He  wishes  to 

rent  the  property  so  that  he  may  have  a  net  income  of  6% 
on  his  investment.  His  yearly  expenses  average  $140.  Find 
the  monthly  rent  Mr.  Hansen  must  receive  from  his  property 
in  order  to  have  a  net  income  of  6%.     (Difficult  for  pupils.) 

2.  Find  the  selling  price  of  a  horse  sold  at  a  loss  of  15%o  which 

amounted  to  $42. 

3.  Given  the  gain  and  the  selling  price,  tell  how  to  find  the  gain 

per  cent  and  illustrate  with  a  problem  of  your  own. 

4.  At  the  close  of  the  year  1913,  there  were  640  pupils  in  a  certain 

school.  How  many  more  pupils  must  be  enrolled  during  1914 
to  have  the  enrollment  at  the  end  of  the  year  1914  25%o  greater 
than  at  the  close  of  the  year  1913? 

5.  When  the  gain  and  the  gain  per  cent  are  given  in  a  problem, 

how  is  the  selling  price  obtained?  Illustrate  with  a  problem 
of  your  own. 

6.  If  the  increase  per  cent  is  required,  what  two  things  must  be 

known?     Illustrate  with  a  problem  of  your  own. 

7.  Mr.  Saunders  paid  $10,000  for  a  piece  of  property.     The  monthly 

rent  from  the  property  amounts  to  $125.  If  the  annual  expenses 
average  $280,  what  is  Mr.  Saunders'  net  rate  of  income  from 
the  property? 

8.  State  and  solve  a  problem  in  which  you  are  required  to  find  the 

interest  due  on  a  sum  of  monej^  borrowed  for  one  year  at  6% 
interest. 
9o  What  per  cent  of  f  of  a  yard  is  f  of  a  yard? 

(29) 


10.  There  were  285  words  in  a  spelling  test.     George  spelled  265 

words  correctly.     Find  his  per  cent  of  error. 

11.  There  were  500  words  in  a  spelling  test.     Albert  missed  82  words. 

What  was  Albert's  per  cent  of  correct  work? 

12.  The  population  of  a  certain  town  was  25,000  in  1900.     In  1910, 

the  population  had  increased  5,500.  The  population  in  1910 
M-as  what  per  cent  of  the  population  in  1900?  What  was  the 
per  cent  of  increase? 

13.  By  selling  a  horse  for  $30  more  than  it  cost,  Mr.   Tompkins 

gained  20%.     Find  the  cost  and  the  selling  price  of  the  horse. 

14.  Mr.  Hamilton  bought  tomatoes  at  the  rate  of  three  cans  for  $.25 

and  sold  the  tomatoes  at  the  rate  of  two  cans  for  $.25.  Find 
the  per  cent  of  gain  or  loss. 

15.  There  are  215  boys  and  289  girls  in  a  certain  school.     What  per 

cent  of  the  pupils  are  boys? 

16.  Mr.  Hooj^er  bought  a  lot  for  $3,000,  upon  which  he  built  a  house 

that  cost  $4,500.  The  yearly  expenses  of  the  property  average 
$320.  Find  the  montlily  rent  Mr.  Hooper  must  receive  from 
his  property  in  order  to  have  a  net  income  of  6%. 

17.  Louise  saves  20%  of  her  yearly  salary  and  spends  $960.     What 

is  Louise's  yearly  income? 

18.  A  public  library  had  7,500  books  in  circulation  during  the  month 

of  July.  This  number  represented  75%  of  all  the  books  in  the 
library.  Find  the  number  of  books  not  in  circulation  during 
the  month  of  July. 

19.  Mr.    Green   receives   an   annual   income   of   $1,200   from   prop- 

erty which  originally  cost  him  $8,500.  If  the  yearly  expenses 
average  $240,  find  Mr.  Green's  net  rate  of  income. 

20.  There  are  520  pupils  in  a  certain  school.     210  of  the  pupils  are 

boys.     What  per  cent  of  the  school  are  girls? 

21.*After  losing  10%  of  its  weight,  a  piece  of  ice  weighed  20  pounds. 
What  was  its  original  weight? 

22. -Mr.  Harris  drew  $350  out  of  the  bank.  This  amount  was  25% 
more  than  the  amount  he  had  left  in  the  bank.  How  much 
had  he  originally  in  the  bank? 

23.*Two  men  invested  $10,000  in  a  grocery  business.  One  man  put 
in  20%  more  than  the  other  man.  Find  the  amount  of  capital 
invested  by  each  of  the  partners. 

24.*A  ring  marked  at  20%  above  cost  is  sold  at  a  discount  of  5%. 
Find  the  gain  or  loss  per  cent. 

25.*Go6ds  marked  15%  above  cost  are  sold  at  a  discount  of  15%. 
Find  the  gain  or  loss  per  cent. 

26.*If  a  merchant  sells  |  of  a  yard  of  silk  for  what  ^  of  a  yard  cost 
him,  find  his  gain  or  loss  per  cent. 

27.*In  1911,  there  were  24,072  girls  attending  high  school  in  Cali- 
fornia.    This  was  18%  more  than  the  number  of  boys  attend- 
ing.    Find  the  number  of  pupils  attending  high  schools  in 
California  during  the  year  1911. 
(30) 


28.*By  selling  a  horse  at  a  profit  of  $25,  Mr.  Lucas  received  115% 
of  the  cost  of  the  horse.  What  was  the  selling  price  of  the 
horse  ? 

29.*If  four  tablets  are  sold  for  the  cost  of  five  tablets,  find  the  gain 
or  loss  per  cent. 

COMMISSION,  INSURANCE,  TAXES,  DISCOUNT,  INTEREST. 

The  material  of  pages  192  through  217  of  the  text  may  be  supple- 
mented by  the  explanations  and  drill  work  of  Bulletin  No.  30.  The 
topics  of  commission,  insurance,  taxes,  etc.,  are  merely  an  application 
of  the  work  of  percentage  to  business  transactions.  Pupils  usually 
fail  to  master  such  topics  because  they  have  no  understanding  of 
business  situations  or  of  the  terms  used  in  the  problems.  Normal 
students  will  be  expected  to  be  familiar  with  any  of  the  problems  of 
the  text  on  pages  192  through  277  as  well  as  all  of  the  work  in  Bulletin 
No.  30.  Students  should  strive  to  obtain  as  full  a  knowledge  as  possible 
of  these  business  transactions. 

Test  on  Commission,  Insurance,  Taxes,  Interest. 

The  following  miscellaneous  review  problems  may  be  used  as  a  guide 
when  it  is  necessary  to  test  a  pupil's  knowledge  of  commission,  insur- 
ance, taxes,  interest  or  discount.  At  the  beginning  of  a  term  or  when 
a  new  pupil  enters  a  class,  such  a  test  is  useful.  Pupils  can  then  be 
drilled  upon  the  points  in  which  their  knowledge  is  deficient. 

1.  What  must  be  given  in  a  problem  in  order  that  you  may  find 

the  rate  of  an  agent's  commission?  Illustrate  by  stating  and 
solving  a  problem  of  your  own. 

2.  What  is  meant  by  the  net  proceeds  of  a  sale?     Illustrate  by  a 

problem  of  your  own. 

3.  When  cartage  and  other  expenses  are  to  be  paid,  is  the  agent's 

commission  figured  on  the  amount  left  after  deducting  expenses  ? 
Give  a  reason  for  your  answer.  Illustrate  by  stating  and  solv- 
ing a  problem  of  your  own. 

4.  Name  as  many  different  occupations  or  situations  as  you  can  in 

which  a  man  receives  a  commission  as  all  or  part  of  his  pay 
for  services  rendered  to  others. 

5.  What  must  be  kno^vn  in  a  problem  in  order  that  you  may  find 

the  amount  of  an  agent's  commission?  Illustrate  by  stating 
and  solving  a  problem  of  your  own. 

6.  If  the  selling  price  and  the  net  proceeds  are  given  in  a  problem, 

tell  how  you  may  find  the  rate  of  the  agent's  commission. 
Illustrate  by  stating  and  solving  a  problem  of  your  own. 

7.  What  is  the  difference  between  the  assessed  value  and  the  real 

value  of  a  piece  of  property?  Illustrate  with  a  problem  of 
your  own. 

8.  What  must  you  know  in  order  to  find  the  rate  of  taxation  which 

will  yield  a  certain  amount  of  revenue?  Illustrate  by  stating 
and  solving  a  problem  of  your  ow^n. 

(31) 


9.  If  the  amount  of  a  man's  tax  bill  and  the  rate  of  taxation  are 
knoAvn.  what  amounts  may  be  found?  Illustrate  by  stating 
and  solving  problems  of  your  own. 

10.  Is  every  property  owner  compelled  to  insure  his  property  against 

loss  by  fire? 

11.  What  must  you  know  in  order  to  find  the  amount  of  a  man's 

tax  bill  ?  Illustrate  by  stating  and  solving  a  problem  of  your 
own. 

12.  In  what  three  ways  may  a  rate  of  taxation  be  expressed  ?     Illus- 

trate by  stating  and  solving  a  problem  of  your  own. 

13.  Name  at  least  five  things  for  which  the  monej''  derived  from  taxes 

is  used. 

14.  a.  Name  some  of  the  items  of  expense  of  the  national  government. 
h.  Name  some  of  the  sources  of  income  of  the  national  gov- 
ernment. 

c.  What  is  the  difference  betw^een  a  specific  duty  and  an  ad 
valorem  duty?  Illustrate  by  stating  and  solving  a  problem 
of  your   owm. 

15.  Is  the  rate  of  taxation  the  same  for  all  taxpayers  who  live  in  a 

certain  citj^,  town  or  district? 

16.  Give  some  reasons  why  two  men  living  in  the  same  block  may 

pay  different  rates  of  insurance. 

17.  Name  at  least  five  things  which  would  raise  the  rate  of  a  man's 

insurance. 

18.  Illustrate  what  is  meant  by  the  face  of  a  policy.     What  is  the 

premium  ? 

19.  How  is  it  possible  for  an  insurance  company  to  make  the  premium 

on  a  three-year  policy  or  a  five-year  policy  cheaper  than  that 
of  an  annual  policy  renewed  each  year  for  the  same  length  of 
time? 

20.  What  must  you  have  given  in  a  problem  in  order  to  find  the  rate 

of  insurance?  Illustrate  by  stating  and  solving  a  problem  of 
your  own. 

21.  What  must  you  have  given  in  a  problem  in  order  to  find  the 

amount  of  the  premium  due  on  a  three-year  policy  ?  Illustrate 
by  stating  and  solving  a  problem  of  your  own. 

22.  Name  at  least  four  important  things  which  should  appear  in  a 

policy. 

23.  If  the  amount  of  premium  and  the  rate  of  insurance  are  given 

in  a  problem,  what  unknown  amount  may  you  find  ?  Illustrate 
by  stating  and  solving  a  problem  of  your  own. 

24.  Mr.  Wilkins'  property  valued  at  $8,000  was  assessed  for  60%  of 

its  value  at  the  rate  of  $2.27  per  $100.  Find  the  amount  of 
Mr.  Wilkins'  tax  bill. 

25.  Mr.  Ruskin  received  $45  for  selling  some  fruit  on  a  commission 

of  3%.     Find  the  net  proceeds  of  the  sale. 

26.  What  is  the  compound  interest  on  $500  borrowed  for  1^  years 

at  the  rate  of  6%  if  the  interest  is  compounded  semi-annually? 
(32) 


r 


27.  What  was  the  total  duty  on  six  dozen  table  knives  worth  $-1.50 

per  dozen  if  the  specific  duty  is  16^  on  each  knife  and  the  ad 
valorem  duty  15%  ? 

28.  A  watch  marked  $45  was  sold  by  a  wholesale  dealer  to  a  retail 

merchant  at  a  discount  of  20%.  The  retail  dealer  sold  the 
watch  at  its  original  marked  price.  Find  the  retailer's  gain 
or  loss  per  cent. 

29.  A  hotel  worth  $50,000  was  insured  in  one  company  for  $12,000, 

in  another  company  for  $15,000  and  in  a  third  company  for 
$1jO,000.  If  the  hotel  was  damaged  to  the  extent  of  $20,000, 
how  much  of  the  damage  would  each  company  pay? 

If  the  hotel  were  damaged  to  the  extent  of  $40,000,  what 
amount  of  the  damage  would  each  company  pay? 

30.  What  is  the  interest  on  $350  borrowed  October  8,  1912,  and 

returned  with  interest  at  5%  on  November  6,  1913?  (Use 
exact  time.) 

31.  Mr.  White's  taxes  amounted  to  $75.     What  was  the  actual  value 

of  his  property  if  the  tax  rate  was  $2.04  per  $100  and  the 
property  was  assessed  for  60%  of  its  real  value  ? 

32.  Mr.  Roberts  insured  his  house  worth  $6,000  for  f  of  its  value 

for  three  years  at  $1.10  per  $100.  What  was  the  amount  of 
his  premium? 

33.  A  real  estate  agent  sold  two  lots  for  $12,500.     He  returned  $8,500 

to  the  owner  of  the  lots.  What  rate  of  commission  did  the 
agent  charge? 

MENSURATION. 

The  work  in  mensuration  on  pages  221  through  252  of  text  may 
be  supplemented  by  the  work  of  Bulletins  Nos.  31  and  32.  Normal 
students  are  expected  to  be  familiar  with  the  work  of  these  bulletins 
as  well  as  the  work  of  above  pages  of  the  text. 

Test  Questions  on  Mensuration. 

The  following  questions  and  problems  are  a  test  of  the  work  of  the 
text,  pages  221  through  252  and  of  Bulletins  Nos.  31  and  32.  Normal 
.students  should  be  familiar  with  the  answers  to  these  questions. 

1.  Explain  with  the  aid  of  a  diagram  why  the  area  of  an  obtuse- 

,    ,  ,   .       ,     .     b  X  a 
angled  triangle  is   — - 

2.  Write  the  formula  for  the  area  of  a  circle  and  for  the  circum- 

ference of  a  circle.  State  and  solve  a  problem  to  illustrate 
each. 

3.  Illustrate  with  a  drawing  the  difference,  if  any.  between  three 

square  inches  and  a  three-inch  square. 

4.  State  and  solve  a  problem  of  your  own  in  which  you  are  required 

to  find  the  area  of  the  entire  surface  of  a  cylindrical  log. 

5.  How  do  you  know  when  to  use  the  square  measure  and  when  to 

use  the  cubic  measure  in  working  with  cylinders  ? 

(33) 


6.  If  the  area  of  a  rectangle  is  given  in  acres  and  the  length  of 

one  side  in  rods,  how  is  the  perimeter  of  the  rectangle  found? 
Illustrate  with  a  problem  of  your  own. 

7.  How  many  acres  are  there  in  a  section  and  one  half  of  land  ? 

8.  Express  in  feet  the  length  and  the  width  of  a  square  field  the 

area  of  which  is  one  acre. 

9.  Write  the  table  of  square  measure. 

10.  What  is  the  perimeter  of  a  field  one  mile  square?     What  is  its 

area? 

11.  If  the  circumference  of  a  circle  is  divided  by  3f  and  that  answer 

by  2,  what  is  the  result  called? 

12.  If  the  area  of  the  floor  of  a  square  room  is  one  square  rod.  what 

is  the  length  and  width  of  the  room  expressed  in  feet  ? 

13.  With  the  aid  of  a  diagram  find  the  area  of  the  largest  circle 

which  may  be  cut  from  a  three-inch  sc|uare. 

14.  Why  is  it  that  the  longest  oblique  line  of  an  obtuse-angled  triangle 

is  not  the  altitude  of  the  triangle? 

15.  If  the  area  of  a  right-angled  triangle  is  270  square  rods,  what  is 

the  area  of  the  rectangle  which  has  the  same  base  and  altitude 
as  the  triangle? 

16.  Write  three  formulas  for  the  area  of  a  triangle.     State  and  solve 

a  problem  to  illustrate  the  advantage  of  each  formula. 

17.  Given  the  circumference  of  a  circle,  how  may  its  radius  be  found  ? 

Illustrate  by  stating  and  solving  a  problem  of  your  own. 

18.  Show   by   a   diagram   the   relation   between   the   diagonal   of   a 

rectangle  and  the  longest  side  of  a  right  triangle. 

19.  Draw  a  trapezoid.     Assign  dimensions  to  it  and  find  its  area  in 

acres. 

20.  Is  the   altitude   of   a   figure   alwaj's   drawn   within   tlie   figure? 

Illustrate. 

21.  What  is  the  name  of  the  solid,  all  dimensions  of  which  are  equal  ? 

22.  Draw  three  different  kinds  of  triangles.     With  dotted  lines  show 

the  parallelogram  of  which  each  triangle  is  one  half.  Label 
the  base  and  altitude  of  both  the  parallelograms  and  triangles. 
Assign  dimensions  to  each  triangle  and  find  its  area  in  acres. 

23.  What  is  the  formula  for  the  volume  of  a  cylinder  ?     Illustrate 

with  a  problem  of  your  own. 

24.  Is  the  perimeter  of  a  triangle,  square,  rectangle,  or  circle  a  line 

or  a  surface?  What  measure  is  used  in  finding  the  perimeter 
of  a  figure? 

25.  Illustrate   what   is   meant    by   one   line   being   perpendicular   to 

another  line. 

26.  Given  the  circumference  of  a  circle,  how  may  its  area  be  found? 

Illustrate  by  stating  and  solving  a  problem  of  your  own. 

27.  Illustrate   with   drawings   the   difference,    if   any,   between   two 

square  inches,  two  inches,  a  two-inch  cube,  and  a  two-inch 
square. 


(34) 


28.  What  is  the  ratio  of  the  diameter  of  a  circle  to   its  circum- 

ference? What  is  the  ratio  of  the  circumference  of  a  circle 
to  its  diameter? 

29.  Draw  as  many  different  parallelograms  as  possible  and  label  the 

base  and  altitude  of  each.  Explain  with  the  aid  of  a  diagram 
why  the  formula  for  the  area  of  a  parallelogram  with  no  right 
angles  is  the  same  as  the  formula  for  the  area  of  a  rectangle. 

30.  What  is  the  perimeter  and  the  area  of  a  six-inch  square  ? 

31.  Tell  for  what  each  of  the  following  numbers  stands  in  the  square 

table :  640,  30^  9,  144,  160,  208i 

32.  Is  it  entirely  correct  to  say  that  a  triangle  is  one  half  of  a  paral- 

lelogram?    If  not,  make  a  correct  statement. 

33.  Write  the  table  of  Linear  Measure. 

34.  Is  the  circumference  of  a  circle  a  surface  or  a  line  ? 

35.  With  a  problem,  illustrate  the  difference  between   finding  the 

surface  and  the  volume  of  a  cylinder. 

36.  Draw  a  cir'^le.     Shade  its  surface  and  label  its  center,  radius  and 

circumference. 

37.  Given  the  circumference  of  the  base  of  a  cylinder,  how  is  the  area 

of  the  base  of  the  cylinder  found?  Illustrate  by  stating  and 
solving  a  problem  of  your  own. 

38.  How  do  you  find  the  area  of  the  convex  surface  of  a  cylinder 

when  the  circumference  of  the  base  of  the  cylinder  and  the 
altitude  of  the  cylinder  are  given?  Illustrate  by  stating  and 
solving  a  problem  of  your  own. 

39.  A  tree  was  broken  by  the  wind  in  such  a  way  that  the  top  of  the 

tree  struck  the  ground  at  a  distance  of  45  feet  from  the  foot 
of  the  tree.  If  the  broken  part  of  the  tree  was  65  feet  long, 
find  the  length  of  the  part  left  standing. 

40.  A  lot  is  125'  X  340'.     How  much  shorter  is  it  to  walk  diagonally 

through  the  lot  than  around  the  two  adjoining  sides  of  the  lot  ? 

41.  Which  field  has  the  greater  perimeter  and  how  much  greater,  a 

rectangular  field  210  feet  by  350  feet  or  a  square  field  of  equal 
area? 

42.  The  area  of  a  square  field  is  8  acres.     What  is  the  perimeter  of 

the  field  expressed  in  feet? 

43.  Which  has  the  greater  area,  a  rectangular  table  top  4'  3"  x  3' 

or  a  circular  table  top  which  has  a  diameter  of  4'  3"?  How 
much  larger  or  smaller  is  the  rectangular  table  top  than  the 
circular  table  top? 

44.  The  perimeter  of  a  square  field  is  320  rods.     At  $6o  an  acre, 

what  is  the  value  of  a  triangular  field  which  has  the  same  base 
and  altitude  as  the  square? 

45.  Mr.  Hamilton  owns  a"  farm  one  eighth  of  a  mile  .square.     IMr. 

White's  farm  has  an  area  of  one  eighth  of  a  square  mile.  What 
is  the  value  of  each  farm  at  $125  an  acre  ? 

46.  If  a  horse  is  tied  to  the  bottom  of  a  stake  by  a  rope  20  feet  long, 

what  is  the  area  of  the  surface  over  which  the  horse  may  graze  ? 
(35) 


47.  How  mauj'  feet  of  picture  molding  are  needed  for  a  room  the 

length  of  which  is  18  feet  and  the  area  of  the  floor  378  square 
feet? 

48.  Ethel  is  embroidering  a  round  centerpiece  for  her  mother  for 

Christmas.  If  the  greatest  width  of  the  centerpiece  is  36 
inches,  how"  many  inches  of  the  outer  edge  of  the  centerpiece 
will  Ethel  have  to  embroider  each  day  in  order  to  finish  the 
outer  edge  of  the  centerpiece  in  8  days?  How  many  square 
feet  of  the  surface  of  a  table  will  be  covered  by  the  centerpiece 
when  finished? 

49.  A  rectangular  field  114  rods  long  has  an  area  of  1^  acres.  '  If 

land  is  worth  $250  an  acre,  what  is  the  value  of  a  triangular 
field  which  has  the  same  base  and  altitude  as  the  rectangle? 

50.  The  length  of  the  upper  base  of  a  trapezoid  is  12  feet  and  the 

lower  base  is  8  feet.  If  the  perpendicular  distance  between  the 
sides  is  4  feet  6  inches,  find  the  number  of  square  feet  on  the 
surface  of  the  trapezoid. 

51.  How  many  revolutions  will  a  28-inch  bicycle  wheel  make  in  trav- 

eling a  mile? 

52.  How  many  cubic  feet  of  marble  are  there  in  a  column  of  marble 

20  feet  long  and  22  inches  in  diameter? 

53.  How  man}'  rods  of  fence  are  needed  to  enclose  a  square  field  the 

area  of  which  is  8  acres? 

54.  At  $.12|  a  scpiare  foot,  find  the  cost  of  laying  a  three-foot  cement 

walk  around  a  circular  flower  bed  30  feet  in  circumference. 

55.  What  is  the  diameter  of  a  circle  which  has  an  area  equal  to  that 

of  a  seven-foot  square? 

56.  The  hypotenuse  of  a  right  triangle  is  70  feet.     The  altitude  of 

the  triangle  is  30  feet.     What  is  the  perimeter  of  the  triangle .' 

57.  What  is  the  length  of  the  longest  stick  which  may  be  placed  within 

a  box  cubical  in  shape,  each  edge  of  which  measures  2  feet  ? 

58.  Express  in  gallons  the  capacity  of  a  covered  cylindrical  tank 

which  is  20  feet  long  and  3  feet  in  diameter. 

59.  What  is  the  area  of  the  entire  outer  surface  of  the  above  tank? 

(Include  cover.) 

60.  Tell  some  method  of  proving  that  the  ratio  of  the  circumference 

of  a  circle  to  its  diameter  is  3i. 

61.  A  cistern  filled  with  water  is  6  feet  in  diameter  and  40  feet  deep. 

How  many  gallons  of  water  does  it  contain? 

62.  If  there  are  31^  gallons  in  a  barrel,  how  many  barrels  of  water 

may  be  filled  from  this  cistern? 

63.  If  the  bottom  and  sides  of  the  interior  of  the  tank  are  cemented. 

find  the  cost  at  $.12  a  square  foot. 

64.  A  marble  column  is  12  feet  high  and  38  inches  in  circumference. 

What  is  the  area  of  the  convex  surface  of  the  column? 

65.  What  is  its  volume  expressed  in  cubic  yards  ? 

66.  What  is  the  length  of  the  diagonal  of  a  square,  each  side  of 

which  is  10  feet? 

(36) 


CLASSIFICATION   OF    PROBLEMS    OF    PAGES    18-160    OF 
ADVANCED  TEXT. 

By  grouping  under  their  proper  headings  all  problems  of  like  types 
which  are  scattered  throughout  the  various  pages  of  the  text,  there  will 
be  found  to  be  in  most  instances,  sufficient  problems  to  teach  a  given 
type. 

Pupils  who  are  accurate  and  understand  their  work  quickly  should 
not  be  required  to  work  as  many  problems  of  a  given  type  as  pupils 
who  are  slow  and  inaccurate.  The  problems  under  each  type  have  been 
divided  into  three  groups— a,  problems  not  marked  with  stars;  &, 
problems  marked  with  one  star;  c,  problems  marked  with  two  stars. 
Pupils  who  work  the  non-starred  problems  correctly  should  be  per- 
mitted to  skip  the  single  and  double  starred  exercises. 

Teach  pupils  to  write  a  statement  telling  what  is  given  and  what  is 
required  to  be  found  in  each  of  the  following  problems : 

Addition  and  Subtraction  of  Integers  and  Decimals. 

Page  19—5-6 ;  *7-8 ;  **9-10-ll.  Page  23—5-7-9 ;  *6-8.  Page  24— 
2-3-7;  *4-5.  Page  25—6;  *5-7.  Page  28— 1-2-3-4-5-6-7.  Page  30— 
4-5-6.     Page  73—5-6-7.     Page  36—1  through  8. 

Multiplication  and  Division  of  Integers  and  Decimals. 
Page  35— Sec.  37,  2-3-4.  Page  41— Sec.  45,  3.  *Page  159—7-8-9-10. 
Page  130—12.  Page  50— Sec.  58,  2-3 ;  *4-5 ;  **6.  Page  50— Sec.  59, 
1-3;  *2-4.  Page  53—3-5-8;  *4-6-7-9.  Page  56— Sec.  67,  1-2;  *3- 
4-5.  Page  56— Sec.  68,  1-4-5 ;  *2-3.  Page  67— Sec.  86,  1-2-4^7.  Page 
72—9.  *Page  67—3-5-6-8.  **Page  129—10-11.  Page  159—1-2-4- 
12-13;  *3-5-14.  Page  56— Sees.  67-68  (entire).  Page  164—6-7-8-9- 
10-11-12-13-14-15-16-23-24. 

Addition  and  Subtraction  of  Fractions. 
Page  98—4-8-9-10.     Page  130—22-23.     Page  110—2-3.     *Primary 
Text  page  188—1-2.    *Page  190—5-6-7-8.    *Page  191—5-6-7.    *Page 
.192—5-6.     **Page  203—1-2-4-5-6.     Page  204—1-2-3. 

Multiplication  and  Division  of  Fractions. 

Page  118—4.  Page  144— Sec.  187,  4-5.  Page  130—25.  Page  127— 
6-7-8-10-11.  Page  127— Sec.  171,  2.  Page  128—8-9-10-11.  Page 
144— Sec.  187,  3-6.     Page  118— Sec.  160,  2-3-5. 

Given  Whole  to  Find  Part. 
Page   57—2-3-4-6-9-11;    *5-7-10-12;    **8-13-14.     Page   58—1-4; 
*2-3 ;  **5-6.     Page  129—6-7.     Page  144— Sec.  187,  1. 

Given  Part  to  Find  Whole. 

Page  59— Sec.  72,  No.  1  through  12.  Page  139—2-^5-7-9-10.  *Page 
129—8-9.  Page  139—3-6-8-11-12-14-15-17-23;  **13-16-18-19-20- 
21-22-24-25-26-27.  Page  141— See.  183,  entire.  Page  143—3-5-6-7- 
8-10;  *l-2-4-9-ll.     Page  144— Sec.  187,  2. 

(37) 


Profit  and  Loss. 

Page  142—14-15-16.     Page  152— Sec.  199. 

Bills  and  Receipts. 
Pages  42-43.     Page  61—1-2.     Bull.  No.  29— Pages  41-42-43-97-98. 
See  Test  Sec.  39  of  this  book. 

Average  and  Ratio. 

Page  41— Sec.  45,  1.     Page  118— Sec.  160,  6.     Page  54—55-145-146. 
Bull.  29— Pages  73-74. 

Area  of  Rectangles. 

The  following  problems  of  the  text  maj^  also  be  used  as  additional 
drill  work  while  teaching  pages  80  to  84: 
Page  130— No.  15-16-17-18. 
Page  128— No.  7-15-17. 
Page  129— No.  1-2-3-4-5. 
Page  165— No.  24. 
See  Test  See.  42  of  this  book. 

Cubic  Measure. 

Pages  8^85-86. 
.     Page  236—6-7. 


(38) 


SAN  FRANCISCO  STATE  NORMAL  SCHOOL  PUBLICATIONS. 

The  Teachers'   Manuals. 

Some  years  ago  the  San  Francisco  State  Normal  School  undertook  the  publication 
of  courses  of  study  for  teaching  the  various  subjects  of  the  curriculum  of  the  elementary 
school.  These  courses  were  prepared  by  members  of  the  faculty  and  were  the  outgrowths 
of  daily  experience  in  directing  and  supervising  the  teaching  by  student  teachers  in  the 
Normal  Elementary  School.  The  original  purpose  of  these  publications  was  to  furnish 
to  these  student  teachers  directions  for  teaching  each  of  the  subjects.  An  essential 
necessity  in  their  construction  was  that  they  should  be  very  explicit,  specific  and 
practicable  in  use.  Gradually  there  grew  a  demand  for  them  by  teachers  in  the  public 
schools,  and  the  Normal  School  began  to  print  larger  editions  in  order  to  fill  this  new 
need.  The  demand  from  the  public  school  sources  has  now  grown  to  such  proportions 
that  one   chief  service  of  the  institution  is   that  of  its  publications. 

Pupils'   Exercise   Books. 

Up  to  1912  the  publications  had  been  confined  to  courses  of  study  for  the  assistance 
of  teachers.  During  1912  the  publication  of  pupils'  exercise  books,  accompanying  the 
teachers'  bulletins,  was  commenced.  In  one  type  of  these  exercise  books  the  pupils 
write  directly  in  printed  lessons.  This  device  saves  a  large  amount  of  labor  and  time 
of  the  teacher  in  copying  upon  the  board  and  in  oral  instructions.  Further,  it  saves  the 
pupils'  time  in  copying  from  the  board.  But  pupils  can  make  progress  two  or  three  times 
faster  than  by  the  usual  method,  and  the  work  is  done  much  more  effectively  and  without 
the  sense  of  drudgery  either  to  pupil  or  teacher.  The  exercise  books  are  printed  upon 
paper  that  will  take  ink.  They  cost  little  or  no  more  than  the  common  blank  books 
of  the  same  quality  of  paper. 

Monographs. 

There  is  now  commenced  a  series  of  monographs  of  a  practical  nature,  aimed  to 
assist  or  suggest  further  development  of  a  greater  efficiency  of  school  instruction. 

Three  Series. 

There  have  been  three  series  of  publications  in  time — one  issued  prior  to  the  great 
fire  of  1906,  of  which  no  numbers  now  remain;  a  series  begun  in  1907  and  continued 
to  1914,  and,  finally,  the  Pupil's  Self-Instruction  Series,  begun  in  1914.  The  latter  two 
will  be  found  listed  below. 

System  of  Publication. 

The  expense  of  these  publications  is  borne  chiefly  by  a  revolving  fund  obtained  by 
their  sale.  They  are  printed  in  the  State  Printing-  Office  and  sold  practically  at  manu- 
facturing cost.  They  are  issued  merely  upon  the  autjriority  of  the  individual  authors  and 
the  editor  of  the  series,  and  do  not  represent  a  general  or  necessarily  permanent  policy 
of  the  school,  nor  a  consensus  of  its  faculty  or  trustees. 

How  to  Order. 

All  orders  must  be  accompanied  by  school  district  warrant  check,  money  order  or 
stamps.  We  cannot  fill  orders  which  require  keeping  of  accounts.  As  most  of  the 
purchases  of  bulletins  and  pupils'  exercise  books  are  now  made  by  the  school  districts, 
teacliers  who  send  orders  should  be  careful  to  secure  the  signature  of  trustees  to  warrants 
in  payment  for  orders,  so  that  delays  may  be  avoided.  Be  careful,  also,  in  filling  out 
orders  that  the  bulletins  are  listed  and  are  not  out  of  print.  We  cannot  exchange 
publications  once  purchased  unless  error  has  been  made  and  the  request  is  made  within 
three  days.  Kindly  avoid,  so  far  as  possible,  conditions  which  require  special  corre- 
spondence  in    the   business   department. 

MONOGRAPHS. 

Monograph  A.  A  remedy  for  Lock-Step  Schooling;  a  preliminary  report  upon  the 
weakness  and  impossibilities  of  the  class  system  of  instruction,  and  progress  to  date  in 
substituting  therefor  an  individual  system  of  teaching.  By  Frederic  Burk.  Issued  free 
upon  application. 

Monograph  B.  Outline  courses  in  general  information  and  general  intelligence.  This 
monograph  undertakes  to  map  out  the  beginning  of  a  reorganization  of  the  high  school 
course  of  study,  not  only  for  the  better  preparation  of  those  intending  to  become  teachers, 
but  also  in  the  general  cause  of  wider  preparation  of  all  students  in  industrial,  civic  and 
social  intelligence.     To  students  intending  to  enter  the  San  Francisco  State  Normal  School 

(39) 


the  monograph  will  be  sent  free.  Section  I  printed  as  separate  pamphlet;  American 
History  and  Civics,  by  P.  F.  Valentine.  Section  II,  pampltlet  for  General  History,  Science, 
and  Literature,  Arithmetic,  Geography,  and  Music.  Section  III,  Spelling,  Language,  and 
Grammar.  Price — the  three  sections  will  be  sent  for  25  cents,  postpaid;  in  lots  of  25  or 
more,  expressage  or  freight  paid  by  purchaser,  7%   cents  per  section. 

Monograph  C.  (In  preparation.)  This  will  be  a  sequel  to  Monograph  A,  giving  the 
data  of  results  of  one  year's  experience  in  operating  an  individual  system  of  instruction 
in  the  Normal  Elementary  Department.  It  will  show  the  records  of  about  500  pupils, 
their  variations  in  rates  of  progress,  etc. 

Monograph  D.  Critical  Difficulties  in  the  Teaching  of  Arithmetic.  For  teachers, 
and  for  students  preparing  for  admission  to  the  Normal  School.  By  Mary  A.  Ward.  Price 
15   cents,   3  cents  added  for  postage. 


PUPIL'S  SELF-INSTRUCTION  SERIES. 

(Adapted  to  an   Individual    Method  of  Teaching.) 

The  occasion  and  general  plan  for  this  series  is  set  forth  in  Monograph  A.  We 
are  at  this  date  (August,  1914)  beginning  the  publication  of  a  series  of  pupils'  exercise 
books  and  teachers'  manuals  adapted  to  use  under  the  individual  system  of  instruction. 
They,  of  course,  may  also  be  used  under  the  class  systein  and  will  assist  in  teaching 
by  the  state  series  texts.  Their  plan  of  construction  embodies  the  features  outlined  in 
Monograph  A — the  "elastic"  lesson,  by  which  the  number  of  exercises  to  secure  compre- 
hension or  accuracy  varies  according  to  individual  need;  the  adaptation  to  simplicity  of 
language;  the  lesson  directions  whereby  the  pupil  can  make  his  own  rate  of  progress  and, 
to  a  large  extent,  independently  of  prescribed  lessons  or  help  from  the  teacher;  the 
cumulative  reviews  by  which  all  principles  once  learned  are  carried  forward  automatically. 

This  series  will  be  found  invaluable,  especially  for  the  rural  school  where  pupils  must 
depend  largely  upon  their  own  resources.  The  teachers'  manuals  will  give  full  directions 
for  operation  of  the  system. 

Prices. 

Except  where  specially  stated,  the  price  of  the  numbers  of  the  Self-Instruction  Series 
will  be  as  follows: 

At  the  Normal  School,  10  cents  each; 

By  mail,  12  cents  each,  postpaid; 

By  express  or  freight,  $7.50  per  hundred,  transportation  charges  paid  by  buyer. 

Arithmetic.     By  Frederic  Burk  and  Mary  A.  Ward. 

No.  20 — Teachers'  Manual  to  accompany  pupils'  books,  Nos.  21,  22,  and  23,  giving 
directions,  answers  to  examples  and  supplementary  examples.  Price — 25  cents  each,  post- 
paid. 

No.  21 — Pupils'  Exercise  Tablet  in  addition  and  subtraction. 

No.  22 — Pupils'  Exercise  Tablet  in  multiplication  and  short  division. 

No.  23 — Pupils'   Exercise  Tablet  in  compound  multiplication  and  long  division. 

No.  29 — Problems  in  Percentage.  Double  number.  Price — 20  cents  each  at  Normal 
School;  24  cents  by  mail;  $15.00  per  hundred  by  express. 

No.  30 — Applications  of  Percentage. 

No.  31 — Problems  in  Mensuration,  Part  I. 

No.  32 — Problems  in  Mensuration,  Part  II. 

Language.     By  A.   S.  Boulware  and  Ethel  G.  Smith. 

No.  42 — Pupils'  Exercise  Tablet  in  Language,  Part  I  (for  fifth  grades). 

No.  43 — Pupils'  Exercise  Tablet  in  Language,  Part  II. 

No.  44 — Pupils'  Exercise  Tablet  in  Language,  Part  III. 
Grammar.     By  Ethel  G.  Smith  and  Frederic  Burk. 

No.  51— Part  I. 

No.  52— Part  II. 

No.  53— Part  III. 

No.  54— Part  IV. 

No.  55 — Part  V.     (In  preparation.) 

No.  56 — Part  VI.     (In  preparation.) 
History.     By  P.  F.  Valentine. 

No.  80 — Pupils'  Exercise  Book,  Part  I,  to  accompany  advanced  state  text.  (Columbus 
through  Jefferson.) 

No.  81 — Pupils'  Exercise  Book,  Part  II,  to  accompany  advanced  state  text.  (Madi- 
son through  Civil  War.) 


(40) 


No.  82 — Pupils'  Exercise  Book,  Part  III,  to  accompany  advanced  state  text.  (Civil 
War  to  present.) 

No.  83— Difficulties  of  liistory  texts  simply  explained.  (Democracy,  the  Constitution, 
Centralized  Government,  Religious  Toleration,  Monroe  Doctrine,  Spoils  System,  Civil 
Service  Reform,   the   Tariff,   etc.) 

Geography.     By  F.   W.   Hoffmann. 

Bulletin  No.  18,  Teachers'  Manual,  with  two  pupils'  exercise  books  in  Map  Geography, 
is  partly  constructed  upon  the  individual  plan.  It  is  already  published  and  may  be 
obtained  upon  application.     (See  next  hst.) 

IN   PREPARATION. 

There  are  in  preparation,  for  publication  during  the  year,  the  following: 

Arithmetic.  Two  tablets  in  formal  arithmetic  for  fractions  and  decimals;  one  exer- 
cise book  in  arithmetic  problems  in  integers;  one  exercise  book  in  denominate  numbers; 
one  exercise  book  in  application  of  percentage. 

Language.     Two  exercise  tablets  for  third  and  fourth  grades,  respectively. 

Phonics.     A  series  of  exercise  books. 

Writing.     A  series  of  exercise  books. 

Drawing.    A  series  of  exercise  books. 

Music.    A  series  of  exercise  books  in  formal  note  work. 


TEACHERS'  MANUALS  AND  PUPILS'  EXERCISE|BOOKS. 

(Series  Published  1907  to  1914.) 

(Out  of  print:  Nos.  1.  2,  3,  5,  6,  7,  8  and  13.  The  materials  of  these  have  largely  been 
absorbed  in  revised  editions  represented  by  the  later  numbers.) 

No.  4 — A  Course  of  Study  in  IVIap  Geography;  paper  bound,  52  pages.  By  Allison 
Ware.     Price — by  mail,  postpaid,  30  cents. 

Outline  Maps — In  connection  with  Bulletin  No.  4,  the  school  publishes  a  series  of  nine 
outline  maps  from  which  pupils  may  trace  outlines  for  use  in  location.  These  maps  are 
9  by  12  inches  in  size.  They  represent  the  following  areas:  North  America,  South 
America,  Europe,  Asia,  Africa,  Australia,  United  States,  California,  and  the  hemispheres. 
Price — by  mail,  postpaid,  for  set  of  nine,  15  cents.     (See  also  Bulletin  No.  18.) 

No.  9 — A  Course  of  Study  in  Language;  174  pages.  By  Effie  Belle  McFadden.  Price — 
by  mail,  postpaid,  30  cents. 

No.  10 — A  Course  of  Study  and  Teachers'  Handbook  in  the  Common  Literature  of 
Life;  207  pages,  paper  bound.     By  Allison  Ware.     Price — postpaid,  40  cents. 

No.  11 — A  Course  of  Study  In  Formal  Arithmetic  and  Teachers'  Handbook.  By  David 
Rhys  Jones.     This  bulletin  is  published  in  various  parts  as  follows: 

Part  I.  Teachers'  Handbook  and  exercises  for  integers;  109  pages,  paper  bound. 
Price — by  mail,   postpaid,   30  cents. 

Part  II.  Teachers'  Handbook  and  exercises  for  common  fractions,  decimals,  per- 
centage, denominate  numbers  and  mensuration;  84  pages,  paper  bound.  Price — by  mail, 
postpaid,   30  f^ents. 

The  Pupils'  Exercise  Books,  Nos.  1,  2,  and  4,  accompanying  the  Handbooks,  are  out 
of  print;  a  limited  stock  of  No.  3  (fractions  and  decimals)  is  still  on  hand.  Price — 10  cents 
by  mail;  the  pupils'  work,  however,  is  included  in  the  Teachers'  Handbook.  Individual 
Series  No.  21  takes  the  place  of  No.  1;  No.  22  and  No.  23  take  the  place  of  No.  2. 

No.  12 — Review  Courses  of  American  History  by  means  of  Composition  Topics,  and 
Teachers'  Handbook  to  the  use  of  the  California  State  Series  Texts.     By  P.  F.  Valentine. 

Part  I.  Teachers'  edition  containing  introduction  and  directions  for  use  of  the  com- 
position method,  the  pupils'  topics  of  the  primary  text,  the  pupils'  topics  of  the  advanced 
text,  a  paragraph  directory  to  the  text,  and  a  cumulative  fact  review  of  the  advanced 
text;  73  pages.     Price — postpaid,  25  cents. 

Part  II.  Pupils'  edition  containing  the  composition  outlines  which  follow  the  state 
primary  text  in  history;  10  pages.  Price — postpaid,  5  cents;  in  lots  of  25  or  more,  freight 
or  expressage  paid  by  purchaser,  4  cents  per  copy. 

Part  III.  Pupils'  edition  containing  the  composition  outlines  which  follow  the  state 
series  advanced  text  in  history,  the  paragraph  directory  to  the  text,  and  the  cumulative 
fact  review  for  the  same;  48  pages.  Price — postpaid,  10  cents;  in  lots  of  25  or  more, 
freight  or  expressage  paid  by  purchaser,  8  cents  per  copy. 

No.  13 — A  Course  of  Study  In  Applied  Problems  In  Arithmetic,  for  Supplementary  Use. 
By  Mary  A.  Ward. 

Out  of  print — order  No.  29,  Self-Instruction  Series. 


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No.  14 — A  Course  of  Study  in  the  Teaching  of  Composition,  Language  and  Spelling; 
paper  bound.  By  Effle  B.  McFadden,  assisted  by  Ethel  G.  Smith.  Teachers'  edition  for 
first  three  years.  Price — postpaid,  25  cents.  Nos.  1  and  2  of  the  pupils'  exercise  books 
heretofore  accompanying  this  handbook  are  out  of  print;  they  are  replaced  by  Nos.  40 
and  41,  Self-Instruction  Series.  A  small  stock  of  Nos.  3  and  4  still  remains.  Price — 10 
cents,  postpaid. 

No.  15 — A  Simplified  Course  of  Study  in  the  Teaching  of  Grammar;  teachers'  edition, 
paper  bound,  187  pages.     By  Frederic  Burk,  Effie  B.  McFadden  and  Irving  Brazier.     Price 

postpaid,  40  cents.     The  pupils'  exercise  books  accompanying  this  handbook  are  all  out 

of  print  and  are  being  replaced  by  Nos.   51,  52,   53,  and  54  of  the   Self-Instruction  Series. 

No.  16 — A  Course  of  Study  in  Phonics.  By  Corrine  H.  Johnstone  and  Frederic  Burk. 
Teachers'   Edition,   Part  I,   90   pages.     Price— postpaid,   20  cents. 

Pupils'  Phonic  Exercise  Book  No.  1  (containing  exercises  reprinted  from  teachers' 
edition).  Price— postpaid,  10  cents;  in  lots  of  25  or  more,  freight  or  expressage  paid  by 
purchaser,  TVz  cents. 

No.  17 — A  Composition  Course  in  American  Government  and  Pupils'  Handbook  to  the 
State  Series  Text  (Dunn's  Community  and  Citizen)  with  Supplement  containing  revised 
or  additional  paragraphs  upon  conservation,  some  California  laws.  Interstate  Commerce 
Commission,  California's  compulsory  education,  direct  election  of  United  States  senators, 
direct  primary,  township  and  county,  the  commission  form  of  city  government,  the 
initiative,  referendum  and  recall,  the  cabinet;  paper  bound,  about  40  pages.  By  P.  F. 
Valentine.  Price — postpaid,  15  cents;  in  lots  of  25  or  more,  expressage  or  freight  paid 
by  purchaser,  10  cents  per  copy. 

No.  18 — A  Course  of  Study  in  Map  Geography;  can  be  used  in  trades  as  low  as  fourth. 
By  F.  W.   Hoffman. 

Teacl 
postpaid. 


Pupil, 
Price — 12   ■ 

Pupilf 
maps  and 
more,  freig 

Outlin 

No.   1,  wor 
postpaid. 

Pupils 
Price — 10  c 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
THIS  BO"^sT^j^ipED  BELOW 


Doks.     Price — 10  cents. 


AN  INITIAL  FINE  OF  25  CENTS 

AJM    1J»J-XX«."  RETURN 

W.UU  BE  ASSESSED   FOR  Fe^T^  ^^^^^ 

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^^V^;r  T^^^-l°ON    -%    SEVENTH     DAV 


pupils  write  directly. 

dual  instruction,  with 
stpaid;  in  lots  of  25  or 
r  both  exercise  books. 
?iven  for  mounting) ; 
Price — 10   cents  each, 


and  California). 


AriORD    BROS. 


)D  2257 


',  ^^ 


i 


